Yes, there are references to Mary in the Old Testament—vastly less than those to the Christ, Jesus, who is the entire focus of the Bible, but nonetheless, there are references.

I will be answering from a Catholic perspective.

The references to Mary in comparison to those of Christ being rather scant, however, says little as to the significance of Mary, as we know that the greatest prophet ever born (except Christ) was John the Baptist (and he is mentioned relatively less than many other prophets or other figures in the Bible):

Luke 7:28

For I say to you: Amongst those that are born of women, there is not a greater prophet than John the Baptist. But he that is the lesser in the kingdom of God, is greater than he.

So even though mentioned relatively little, and even then only in terms of Christ, and not independantly, this does not indicate a lack of importance (Luke 7:26). So it is with Mary also. The Bible is about Christ, and Mary is not its focus. Nonetheless, we find references to Mary in both the Old and New Testaments, some of which are very significant indeed!

Kinds of Prophecy

First, it's helpful to distinguish the various kinds of references to future events or figures in the Old Testament. These generally fall under type or prophecy. Often they are merged into a typological, mystical prophecy involving many elements (most of them unconcerned with the chronology of the prophesied events or figures).

A type is an Old Testament figure or event which, while very much real, nonetheless also, ordered in God's inscrutable wisdom and providence, simultaneously foreshadowed things to come—something greater in the New Testament, superior to its shadow in the same way a shadow is to the one casting it (which is always the casewith types, by the way):

Colossians 2:17

These are a shadow of things to come, but the body [that casts it] is of Christ.

Examples of types are too many too count or recount, but a few are: 1 Peter 3:18-22 (the flood waters/waters of baptism); John 3:14/Numbers 21:8-9 (a sign to look unto to be saved/Christ looked to for salvation); *1 Corinthians 5:7 (the Paschal Lamb/Christ); Genesis 22:12,8; John 3:16 (an only son being sacrificed in obedience) etc.

And prophecy everyone is familiar with; but briefly, they can be either mystical/veiled, or more explicit. An example of a more explicit prophecy is Genesis 49:10 or Deuteronomy 18:15. An example of a more mystical prophecy is Zechariah 3:1-8.

Surprisingly to some, most 'prophecies' in the New Testament are actually of the 'type' kind—'look how God foreshadowed the New Covenant'. That is to say, direct, explicit prophecies are really quite rare in comparison to typological prophecy.

For example, "Out of Egypt I have called my son" originally referred to Israel, God's 'son'. Not Jesus. Instead of recognizing or even being aware fo the existence of typology, critics of the New Testament claim the writers in fact didn't know what they were quoting, and used quotes out of context. This is so utterly false, and obviously wrong. St. Matthew was able to read the first half of a sentence (Hosea 11:1). He using typology.

Matthew 2:15

That it might be fulfilled which the Lord spoke by the prophet, saying: Out of Egypt have I called my son.

It doesn't say it fulfilled any prediction or prophecy. He is saying it happened to be a prophetic saying; and Israel had significance beyond the material—as a type of Christ. For example, Christ didn't think 'the Mosaic Law' was a prophecy predicting something with precise dates when He said what He said in Matthew 5:17. Inasmuch as He saw it, like many Old Testament events, to be something recorded so that we would relate to their fulfillment; and pointed to the fact that the Mosaic Law pointed to something outside itself, even the source of grace of which it was a mere channel,and an acquaintance with Christ who was that source, and His higher calling in the New Covenant.

So since the most common kind of prophecy in the New Testament is type, perhaps Mary has many types of her? Well, we'll see that she does, but first, the prophecies.

Since these prophecies are so packed, though they are few, I can only include two, of the respective kinds, in this answer (even then, I will only be touching on them).

Prophecies about and Types of Mary

A New Adam and a New Woman for a New Creation: a Prophecy of Mary

2 Corinthians 5:17. Galatians 4:3-5. 1 Corinthians 15:22.

One of the earliest prophecies in all of Scripture, if not the first prophecy per se, is that of Genesis 3:15, often called the Proto-Gospel (or the first promise of hope since the Fall). After the Fall, when Eve gave into the devil in eating the forbidden fruit, and Adam likewise gave into his wife, God says the following to the serpent—a guise of the devil:

I will put enmity between thee and the woman, and thy seed and her seed: [the woman/her seed] shall crush thy head, and thou shalt lie in wait for [her/the seed's] heel.

Ignoring the latter part, and the translation issues related thereto, we see God promises that the manner in which the devil brought sin into the world (tempting a virgin to disobey God) would be the manner in which God would bring his defeat, crushing his pride (of which the crushing of the "head" is symbolic). Namely, through another 'woman'—a New 'Eve'. Eve gave in; the New Eve will markedly be the opposite. In this is the devil's defeat brought about. What, then, do we find about the Incarnation, the inital and principle fulfillment of this prophecy?

Not an evil or fallen angel causing a woman to disobey God and bring about the Fall, but a good angel, Gabriel, asking a woman to be obedient to His plan to bring about the Redemption. Even before we go further, her primary 'seed' Christ is then the prime candidate, without peeking at the New Testament yet, for a New Adam in this future 'new creation.' (We ultimately are proven to be correct. 1 Corinthians 15:22, 45; Romans 5:12,14,21 etc.)

Luke 1:38

And Mary said: Behold the handmaid of the Lord; be it done to me according to thy word. And the angel departed from her.

This woman appears again at enmity with the devil in the Book of Revelation, in which her seed is identified as Jesus Christ, but also His followers, His disciples, those who follow Him.

This time, she is inserted into a symbolic-narrative vision, yet it's clear that the woman of Genesis 3 is here in view, complete with enmity with a serpent, and her seed Revelation 12:

And a great sign appeared in heaven: A woman ... and behold a great red dragon, ... and the dragon stood before the woman who was ready to be delivered; that, when she should be delivered, he might devour her son. And she brought forth a man child, who was to rule all nations with an iron rod [Jesus]: and her son was taken up to God, and to his throne [the Ascension]. ... And there was a great battle in heaven, Michael and his angels fought with the dragon, and the dragon fought and his angels: ...  And they prevailed not, neither was their place found any more in heaven. And that great dragon was cast out, that old serpent, who is called the devil and Satan, who seduceth the whole world; and he was cast unto the earth, and his angels were thrown down with him ... the devil is come down unto you, having great wrath, knowing that he hath but a short time. ... And the dragon was angry against the woman: and went to make war with the rest of her seed, who keep the commandments of God, and have the testimony of Jesus Christ. ...

Not only is this Mary because her Son is so explicitly Jesus Christ, but because in John 19:27, Christ in His last moments gives His mother, Mary, to be the mystical "mother" of His beloved disciple John, who also wrote Revelation!

Mary's primary 'seed' is Christ, and "the rest of her seed" is His followers: Mary is our spiritual mother (Judges 5:7), as Christ is our spiritual brother! Consider how the two respective 'Adams' treat the respective 'Woman':

Genesis 3:20

And Adam called the name of [the woman] Eve: because she was the mother of all the living.

John 19:26

When Jesus therefore had seen his mother and the disciple standing whom he loved, he saith to his mother: Woman, behold thy son. After that, he saith to the disciple: Behold thy mother. And from that hour, the disciple took her to his own.

The similarity between Eve and the New Eve doesn't stop at their perogatives (as mother of fleshly and spiritual offspring, respectively), or how they were interacted with by angels to bring about the Fall and the Redemption, respectively, of the human race. We see even more parallels. Even in the very way the respective Adam's bring about the respective Fall and Redemption is very similar in connection with the respective 'Eves'.

• Evil angel, Satan, tempts Eve to disobey God's decree; woman tempts Adam with what will bring about the Fall; Adam knows but gives in.

• Good angel, Gabriel, asks Mary to obey God's plan; Mary 'tempts' the New Adam to begin His ministry and thus the suffering of His Passion, and ultimately the Redemption; He knows, but gives in:

John 2:1-5

And the third day, there was a marriage in Cana of Galilee: and the mother of Jesus was there. And Jesus also was invited, and his disciples, to the marriage. And the wine failing, the mother of Jesus saith to him: They have no wine. And Jesus saith to her: Woman, what is that to me and to thee? my hour is not yet come. His mother saith to the waiters: Whatsoever he shall say to you, do ye.

A few things are to be noted here. Jesus gives in, even though His hour is not yet come. He knows the consequences of becoming enemy number one, suffering, being murdered, but 'gives in.' The words 'what to me to thee?' in the Greek are not to be taken for rebuke, since Mary goes on to act as though this means 'OK' (to simplify things). He knows what it entails, as Adam knew that he would "surely die", but nonetheless gave in to the New Eve.

Remarkable parallels! Even more striking, and even further proof that this parallel is what in fact is being conveyed, St. John in this early part of His book is calling to mind the beginning of Genesis throughout: "In the beginning..God...all things were made by him..on the first day...on the second day...on the third day..." etc. What is Jesus' first act of the 'New Creation'? He takes six water pots representing the six days of the first creation, and peforms His first (public?) miracle. He calls to mind, too, the fact that "Mary the mother of Jesus was there" first, and then Jesus. This is, I think, significant. He is definitely calling to mind something significant about Mary and Jesus in this scene.

The Ark of the Covenant : a Type of Mary

One of the most striking types of Mary is the Ark of the Covenant. Looking at this image, you might see why. Take a close look at what this sacred vessel was to built to contain.

Hebrews 9:4

...the ark of the covenant covered about on every part with gold, in which was a golden pot that had manna [which fell from heaven], and the rod of Aaron [the highpriest, and the founder of Israel's priesthood], that had blossomed, and the tables of the testament [word of God on stone].

These were of course symbols of Christ: the true manna which came from heaven (John 6:25-42), the ultimate Highpriest (Hebrews 4:14-16) and the Word of God made flesh (John 1:14), instead of on stone tablets.

Who or what does the container made for Jesus symbolize, then?—it's the same woman "clothed with the sun," or here, gold. (Revlation 11;19; 12:1! cf. Ps 45:9; Mk 10:40; 1 Kings 2:19). Mary is the New Ark. Maybe that's why the old one had to give way to the new (Jeremiah 3:16).

Still not convinced? See how St. Luke quite deliberately, under inspiration of the Holy Ghost, records the similarities between the ark and Mary!

Luke 1:39-56

And Mary rising up in those days, went into the hill country with haste into a city of Judea. And she entered into the house of Zachariah, and saluted Elizabeth. And it came to pass, that when Elizabeth heard the salutation of Mary, the infant leaped in her womb. And Elizabeth was filled with the Holy Ghost: And she cried out with a loud voice, and said: Blessed art thou among women, and blessed is the fruit of thy womb. And whence is this to me, that the mother of my Lord should come to me? For behold as soon as the voice of thy salutation sounded in my ears, the infant in my womb leaped for joy. And blessed art thou that hast believed, because those things shall be accomplished that were spoken to thee by the Lord. ... And Mary abode with her about three months; and she returned to her own house.

Compare the striking parallels with what happens in regard to Mary, and what happens in regard to the Ark of the Covenant—not scattered through its history or Scripture, but in the one scene, even in the one chapter!

2 Samuel 6:1-17

And David again gathered together all the chosen men of Israel, thirty thousand. And David arose and went, with all the people that were with him of the men of Judea to fetch the ark of God, upon which the name of the Lord of hosts is invoked, who sitteth over it upon the cherubims. ... And when they had taken it out of the house of Abinadab, who was in Gabaa, Ahio having care of the ark of God went before the ark. But David and all Israel played before the Lord on all manner of instruments made of wood, on harps and lutes and timbrels and cornets and cymbals. And when they came to the floor of Nachon, Uzzah put forth his hand to the ark of God, and took hold of it: because the oxen kicked and made it lean aside. And the indignation of the Lord was enkindled against Uzzah, and he struck him for his rashness: and he died* there before the ark of God. ... And David was afraid of the Lord that day, saying: How shall the ark of the Lord come to me? And he would not have the ark of the Lord brought in to himself into the city of David: but he caused it to be carried into the house of Obededom the Gethite. And the ark of the Lord abode in the house of Obededom the Gethite three months: and the Lord blessed Obededom, and all his household. ... And David and all the house of Israel brought the ark of the of the Lord with joyful shouting, and with sound of trumpet. And when the ark of the Lord was come into the city of David, Michol the daughter of Saul, looking out through a window, saw king David leaping and dancing before the Lord...

* ^{I've highlighted this to show how God regards His Ark, and how it is unseemly that any man would touch Mary sexually, who bore the Holy One and "God...made flesh," (John 1:1,14) and that she remained a virgin as Christians always taught: even though married to Joseph she asks Gabriel, "How shall this [conception] be, since I know not man?" (Luke 1:34) (a euphemism for 'do not have sexual intercourse').}

One could develop this further. There are other strong types of Mary, and they are all linked. But these are a taster.

How about just one more...

Judith, proto-fulfillment of the Woman of Genesis 3—and type of Mary.

Judith 13:14-26

And it came to pass, when the men had heard her voice, that they called the ancients of the city. And all ran to meet her from the least to the greatest: for they now had no hopes that she would come. And lighting up lights they all gathered round about her: and she went up to a higher place, and commanded silence to be made. And when all had held their peace, Judith said: Praise ye the Lord our God, who hath not forsaken them that hope in him. And by me his handmaid he hath fulfilled his mercy, which he promised to the house of Israel: and he hath killed the enemy of his people by my hand this night. Then she brought forth the head of Holofernes out of the bag, and shewed it them, saying: Behold the head of Holofernes the general of the army of the Assyrians, and behold his canopy, wherein he lay in his drunkenness, where the Lord our God slew him by the hand of a woman. But as the same Lord liveth, his angel hath been my keeper both going hence, and abiding there, and returning from thence hither: and the Lord hath not suffered me his handmaid to be defiled, but hath brought me back to you without pollution of sin, rejoicing for his victory, for my escape, and for your deliverance. Give all of you glory to him, because he is good, because his mercy endureth for ever. And they all adored the Lord, and said to her: The Lord hath blessed thee by his power, because by thee he hath brought our enemies to nought. And Ozias the prince of the people of Israel, said to her: Blessed art thou, O daughter, by the Lord the most high God, above all women upon the earth. Blessed be the Lord who made heaven and earth, who hath directed thee to the cutting off the head of the prince of our enemies. Because he hath so magnified thy name this day, that thy praise shall not depart out of the mouth of men who shall be mindful of the power of the Lord for ever, for that thou hast not spared thy life, by reason of the distress and tribulation of thy people, but hast prevented our ruin in the presence of our God. And all the people said: So be it, so be it.

The typology is strong! Not only does this fulfill in a lesser more material sense Genesis 3:15 but again St. Luke ('Mary's Gospel') we see striking parallels between this and Luke 1 (26-55):

And in the sixth month, the angel Gabriel was sent from God into a city of Galilee, called Nazareth, To a virgin espoused to a man whose name was Joseph, of the house of David; and the virgin's name was Mary. And the angel being come in, said unto her: Hail, full of grace, the Lord is with thee: blessed art thou among women. Who having heard, was troubled at his saying, and thought with herself what manner of salutation this should be. And the angel said to her: Fear not, Mary, for thou hast found grace with God. Behold thou shalt conceive in thy womb, and shalt bring forth a son; and thou shalt call his name Jesus. He shall be great, and shall be called the Son of the most High; and the Lord God shall give unto him the throne of David his father; and he shall reign in the house of Jacob for ever. And of his kingdom there shall be no end. And Mary said to the angel: How shall this be done, because I know not man? And the angel answering, said to her: The Holy Ghost shall come upon thee, and the power of the most High shall overshadow thee. And therefore also the Holy, which shall be born of thee, shall be called the Son of God. And behold thy cousin Elizabeth, she also hath conceived a son in her old age; and this is the sixth month with her that is called barren: Because no word shall be impossible with God. And Mary said: Behold the handmaid of the Lord; be it done to me according to thy word. And the angel departed from her.

And Mary rising up in those days, went into the hill country with haste into a city of Juda. And she entered into the house of Zachariah, and saluted Elizabeth. And it came to pass, that when Elizabeth heard the salutation of Mary, the infant leaped in her womb. And Elizabeth was filled with the Holy Ghost: And she cried out with a loud voice, and said: Blessed art thou among women, and blessed is the fruit of thy womb. And whence is this to me, that the mother of my Lord should come to me? For behold as soon as the voice of thy salutation sounded in my ears, the infant in my womb leaped for joy. And blessed art thou that hast believed, because those things shall be accomplished that were spoken to thee by the Lord. And Mary said:

My soul doth magnify the Lord. And my spirit hath rejoiced in God my Saviour. Because he hath regarded the humility of his handmaid; for behold from henceforth all generations shall call me blessed. Because he that is mighty, hath done great things to me; and holy is his name. And his mercy is from generation unto generations, to them that fear him. He hath shewed might in his arm: he hath scattered the proud in the conceit of their heart. He hath put down the mighty from their seat, and hath exalted the humble. He hath filled the hungry with good things; and the rich he hath sent empty away. He hath received Israel his servant, being mindful of his mercy: As he spoke to our fathers, to Abraham and to his seed for ever.

Mary has such enmity with the devil (as well as those redeemed by Christ as she herself was—"and the rest of her seed") because she was preserved from from ever being prey to original sin—to his snare and poison.

Hence all the types of Judith being 'undefiled' by the enemy and with 'sin', and the ark being made of 'pure gold' to be fit for the presence of the glory of God, being 'blessed of all women', and 'full of grace' before the Incarnation, in preparation for it, and because of its merits: the Redemption of Jesus Christ. This is why 'all generations call her blessed', as the New Eve who did not partake in the fallen creation inasmuch as she was the New Eve who was to be the beginning of the doing-away-with of it. This is perhaps—God knows—why shes says "He who is might has done great things to me." And why unlike anyone else ever, an angel greets a human in this way:

"Hail, full of grace! The Lord is with thee: blessed art thou among women."

Maybe this is why he basically renames her, or addresses her as, kecharitomene (full of grace) and doesn't even call her by her name. Because she was the first to be reinstated with the original grace and justice Adam and Eve were made to have, as the New Eve.

There is much to be said (and speculated) about Mary, the mother of God the Word made flesh, but this is just a small taster.

Silent little Mary God has hidden as a kind of treasure in Scripture, and she is content to keep relatively silent and give all glory, as a model Christian and follower of Jesus, to her Son. Remaining in various (and many!) types and prefigurements and subtle prophecies throughout Scripture, of which you've seen but two touched upon.