Luke 1:31 "And, behold, thou shalt conceive in thy womb, and bring forth a son, and shalt call his name JESUS".

Luke 1:35 "And the angel answered and said unto her, The Holy Ghost shall come upon thee, and the power of the Highest shall overshadow thee: therefore also that holy thing which shall be born of thee shall be called the Son of God"

However, giving Mary a special title like Mother of God was discouraged by Jesus in Luke 11:27-28 "And it came to pass, as he spake these things, a certain woman of the company lifted up her voice, and said unto him, Blessed is the womb that bare thee, and the paps which thou hast sucked. 28 But he said, Yea rather, blessed are they that hear the word of God, and keep it".

Mary was just an ordinary woman.

The title "Mother of God" or Theotokos was first challenged by Nestorius and led to the council of Ephesus as mentioned in an answer above. I will try to summarize the arguments that St. John Cassian gives from scripture in his work Against Nestorius which was written leading up to that council. These arguments are likely common to Orthodox, Catholics, and most Protestants.

I will quote the scriptures used and try to paraphrase the arguments as best I can for brevity.

Book II

Luke 2:11

There is born to you this day in the city of David a Savior who is Christ the Lord

The logical extension made is that only God can save and only God is Lord so the verse effectively says God is becoming incarnate and being born of the virgin.

Luke 1:35

The Holy Ghost shall come upon the and the power of the Most High shall overshadow thee: therefore that holy thing which shall be born of you shall be called the Son of God"

He notes here that only if a being truly holy and divine were born would it be necessary for the virgin to be prepared in such a way. If Mary only conceived of Christ's humanity why such a powerful overshadowing and preparation? This is also a powerful case for Trinitarian theology as we see all 3 persons participating in the incarnation.

Isaiah 7:14

Behold a virgin shall conceive and bear a Son, and they shall call his name Emmanuel, which is interpreted God with us

Cassian argues that this verse clearly teaches that the virgin gives birth to his divinity, not just his humanity.

Titus 2:11-13

The grace of God and our Savior Lord Jesus Christ appeared unto all men ... looking for the blessed hope and coming of the glory of the great God and our Savior Jesus Christ

Here Paul clearly applies the title Lord, Savior and God all to the person of Jesus Christ, whom Mary gave birth to. Because it is clear that Paul is not teaching that a man was born and then God indwelt him, but that the divinity was present at his appearing (i.e. - birth).

After this St John goes on to make comments about how Christ is called both the Son of man and the Son of God and that these titles are applied to the same person. Son of man because he assumed human nature and Son of God because he retained his divine nature and that the angel specifically says that that which is born of her shall be called the Son of God. Therefore calling Mary the Theotokos preserves this Christological orthodoxy.

Book III

Here St. John notes that some people argue that in Exodus 7:1 and Psalm 81 (or 82): 6 that men are also called "god". However, he simply replies that these passages clearly show God bestowing the title of god upon man and not that man is inherently that thing. Which is much different than the passages that speak of Christ.

Galatians 1:1

Now we no longer know Christ according to the flesh. Not of men, neither by man, but by Jesus Christ

So clearly here the title of Christ and Jesus are inseparable from God incarnate as Nestorius wished to make them by calling Mary "Christotokos" and not "Theotokos"

I haven't finished reading the book. But I hope I've given a decent intro to his arguments for the title Theotokos.

In his answer to the same question, Thunderforge asserted,

So it’s clear from the Catholic Church’s lack of scripture on these points in their official doctrines that there is no biblical basis for Mary having the title “Mother of God.”

St. Thomas Aquinas discussed a similar objection in his Summa Theologica, writing,¹

Concerning the fourth [article], it is thus advanced: It seems that the Blessed Virgin should not be called “the Mother of God,” for nothing must be said concerning the Divine Mysteries that is not taken from Sacred Scripture. But nowhere in the Sacred Scripture is it read that she is the mother or parent of God, but that she is the mother of Christ or the mother of the child, as evident from Matt. 1:18. Therefore, it should not be said that the Blessed Virgin is the Mother of God.

Ad quartum sic proceditur. Videtur quod beata virgo non debeat dici mater Dei. Non enim est dicendum circa divina mysteria nisi quod ex sacra Scriptura habetur. Sed nunquam in sacra Scriptura legitur quod sit mater aut genitrix Dei, sed quod sit mater Christi, aut mater pueri, ut patet Matth. I. Ergo non est dicendum quod beata virgo sit mater Dei.

The Latin phrase «quod ex sacra Scriptura habetur»—“that is taken from Sacred Scripture” (note the absence of the negative conjunction nisi)—refers to a belief or dogma that is explicitly stated in scripture. For example, we can say that the belief of the resurrection of the Lord Jesus Christ is “taken from Sacred Scripture” since the Bible explicitly states that “Jesus died and resurrected.”³ Consequently, we may also affirm that such a belief has a biblical basis.

However, not all beliefs with a biblical basis are derived from explicit statements in Sacred Scripture. That is, not all beliefs with a biblical basis are “taken from Sacred Scripture” (ex sacra Scriptura habetur). Rather, some beliefs are derived from Sacred Scripture via logical deductions and implicit statements. One of the most famous beliefs is the doctrine of the Trinity. Nowhere in the Bible does the Bible explicitly mention the word “Trinity,” yet no Protestant would dare say the Trinity lacks a biblical basis.

In his reply to the aforementioned objection, St. Aquinas concedes that the belief that the Blessed Virgin Mary is the mother of God is not explicit (“expressly stated in Scripture”):⁴

...although it is not found expressly stated in Scripture that the Blessed Virgin is the Mother of God...

...licet non inveniatur expresse in Scriptura dictum quod beata virgo sit mater Dei...

But St. Aquinas continues his response to the objection:⁵

...yet it is found expressly [stated] in Scripture that Jesus Christ is true God, as evident in 1 John 5:20,⁶ and that the Blessed Virgin is the mother of Jesus Christ, as evident in Matt. 1:18. Therefore, from the words of Scripture it follows of necessity that she is the Mother of God.

...invenitur tamen expresse in Scriptura quod Iesus Christus est verus Deus, ut patet I Ioan. ult.; et quod beata virgo est mater Iesu Christi, ut patet Matth. I. Unde sequitur ex necessitate ex verbis Scripturae quod sit mater Dei.

St. Aquinas writes «sequitur ex necessitate»—“it follows of necessity.” What “necessity” is he referring to? It is the logical necessity. Gustavus Fischer wrote the following on the phrase necesse est:⁷

Likewise, the verb sequitur is used in syntax concerning logical conclusions.⁸

In other words, although the belief that the Blessed Virgin Mary is the “mother of God” is not explicitly stated in scripture, it is implied by several scriptures and also logically derived by means of a syllogism:

1. Mary is the mother of the Lord Jesus Christ.
2. The Lord Jesus Christ is God.
3. Mary is the mother of God.

To deny that Mary is the mother of God, one must deny that (1) Mary is the mother of the Lord Jesus Christ and/or that (2) the Lord Jesus Christ is God, both of which are explicitly affirmed in the New Testament. Thus, it is demonstrated that the biblical basis of something not explicitly stated in scripture can be established from other explicit statements.

Objection–“Mother of God” suggests “Mother of Trinity”

St. Aquinas does discuss a few other notable objections to the belief that Mary is the mother of God, some of which have been discussed in the comments.

For example, it is a common misconception by some non-Catholics that “mother of God” indicates that the Blessed Virgin Mary is the mother of God the Father and/or the Holy Trinity.

St. Aquinas responds to that objection, writing,⁹

Concerning the third objection, it must be said that, although this name “God” is common to the three persons, yet sometimes it applies only for the person of the Father, sometimes only for the person of the Son or of the Holy Spirit, as it is maintained above (III:16:1; I:39:4). So that when we say, “The Blessed Virgin is the Mother of God,” this name “God” applies only for the incarnate person of the Son.

Ad tertium dicendum quod hoc nomen Deus, quamvis sit commune tribus personis, tamen quandoque supponit pro sola persona patris, quandoque pro sola persona filii vel spiritus sancti, ut supra habitum est. Et ita, cum dicitur, beata virgo est mater Dei hoc nomen Deus supponit pro sola persona filii incarnata.

Emphatically and succinctly stated, the term “mother of God” is a shorthand expression for “mother of God the Son” and must not be understood in any other way.

Objection—“Mother of God” is not an accurate translation of ἡ Θεοτόκος

At face value, this is a valid objection, since the word Θεοτόκος literally means “God-bearer” or “God-birther.” However, it simply needs to be realized that a woman who gives birth to a child is also the mother of that same child. Birthing and motherhood are inseparable.

St. Aquinas wrote,¹⁰

And some woman is called some man’s mother from this: because she conceived him and gave birth [to him].

Ex hoc autem dicitur aliqua mulier alicuius mater, quod eum concepit et genuit.

References

Aquinas, Thomas. Summa Theologica. Vol. 6. Paris: Bloud, 1880.

Aquinas, Thomas. The Summa Theologica of St. Thomas Aquinas. Trans. Fathers of the English Dominican Province. Vol. 16. New York: Benziger, 1911–1912.

Fischer, Gustavus. Latin Grammar together with a Systematic Treatment of Latin Composition. Part Second. New York: Schermerhorn, 1876.

Lewis, Charlton T.; Short, Charles. Harper’s Latin Dictionary: A New Latin Dictionary Founded on the Translation of Freund’s Latin-German Lexicon. New York: American Book, 1879.

Footnotes

¹ ST III, Q 35, A 1, arg. 1
² ST III, Q 35, A 1, ad. 1
³ e.g., 1 Thes. 4:14
⁴ ST III, Q 35, A 4, ad. 1. The Latin word licet followed by the subjunctive verb inveniatur indicates the commencement of a concessive clause.
⁵ ibid.
⁶ Although the referent of 1 John 5:20 is debated, other scriptures clearly affirm that the Lord Jesus Christ is God, e.g., Phil. 2:6–9; John 1:1–14; Col. 2:9; etc.
⁷ p. 141; necesse is the adjective; necessitas is the noun.
⁸ p. 1677
⁹ ST III, Q 35, A 4, ad. 3
¹⁰ ST III, Q 35, A 4, co.

The most sure fact about Mary from Holy Scripture is that she is the Mother of God.

A Fundamentally Christological Doctrine

First of all, what does the honorific 'Mother of God' mean?

Simply put, it is the answer to the fundamental question: Of whom is Mary the mother? And the answer is intrinsically Christological. Jesus Christ is no less than God at any time whatever. There is no such person as a non-divine, non-fully-God, Jesus Christ.

It's no surprise that the truth of Mary's being the Mother of God, and not merely of 'Christ' (as if there is some rogue human-only Christ somewhere) being defined as a dogma of the faith came about due to its denial by a heretic positing a false Christology.

The heretic Nestorius, who divided the natures of Christ to such a degree, or rather made a distinction beyond what is orthodox, and in so doing, knowingly or not, found 'two Christs' in Christ. Where you could say one thing about Christ, and then another, and have them refer, in all actuality, to two different persons, a kind of divine and a human Jesus, which must be distinguished between. An awkward perversion of the hypostatic union, whereby the Christian believes Christ has two natures perfectly united, although not confused, in one Person: Jesus Christ, God the Word incarnate.

The heresy is summarized by the Ecumenical Second Council of Constantinople's condemnation of it in A.D. 533 (Sentence Against the Three Chapters of Nestorius):

The holy synod of Ephesus… has pronounced sentence against the heresy of Nestorius… and all those who might later ... adopt the same opinions as he held ... They express these falsehoods against the true dogmas of the Church, offering worship to two Sons, trying to divide that which cannot be divided, and introducing to both heaven and earth the offense of the worship of man. But the sacred band of heavenly spirits worship along with us only one Lord Jesus Christ.

Even earlier than this, and directly addressing a specific heretical product of his false Christology (that Mary was only the mother of a human nature, and not of the divine Person of God the Word), the Ecumenical Council of Ephesus (A.D. 431) issues the following anathema (Session I, Canon 1):

If anyone will not confess that the Emmanuel is very God, and that therefore the Holy Virgin is the Mother of God [Θεοτόκος], inasmuch as in the flesh she bore the Word of God made flesh [as it is written, "The Word was made flesh"] let him be anathema.

It's pretty simple: Mary's Son was never anyone 'only human' or who had 'only a human nature'. This Jesus never existed. The only Jesus the orthodox Church ever worshiped (else it would indeed be worship of man) the God-man, God the Word incarnate, Jesus Christ, true God and true man. That Mary, therefore, is the Mother of Him is all that is to be proven from Scripture. And that it proves it, is beyond any reasonable dispute whatsoever.

The Testimony of the Holy Ghost

If the Holy Ghost is not allowed to declare that Mary is the Mother of Jesus Christ, or the Mother of God, then no one is. However, He does so.

For in the use of so many of His instruments, he conveys this truth, not as from the point of view of a heretic, or as relating a false opinion, but in His teaching the people of God the Gospel, He freely teaches and instructs us that Mary is truly the Mother of God: that is, the Mother of God the Word.

That Jesus is and always has been God is laughably easy to show from Scripture, and is uncontestable. For which reason we shall assume it.

Scriptures Teaches us Mary is the Mother of Jesus

Through His Evangelists:

Matthew 1:18; 2:14; 2:21; 12:46. Mark 3:31. Luke 2:33,34; 2:48; 2:51; 8:19. John 2:1,3 “the mother of Jesus;” 2:5; 2:12; 19:25,26.

Angels: Matthew 2:13; 2:20.

Men: Matthew 12:46; 13:55; Luke 18:20.

People under His express and direct inspiration: Luke 1:43.

St. Luke recounting the Acts of the Apostles: Acts 1:14 “the mother of Jesus.”

So Mary is really the Mother of “God...the Word...made flesh.” (Galatians 4:4; John 1:1,14; Revelation 19:13 cf. Isaiah 63:1-3; Genesis 49:10-11).

'Mother of God'

Our title 'Mother of God' and 'Theotokos', which are synonymous, are synonyms for 'the Mother of the Lord.' Because the Lord Jesus was never only Lord, but God. And the Holy Spirit tells us that Mary is “the Mother of [the] Lord” most explicitly.

Mary is the Mother of He who is “Lord..and..God,” mind you (John 20:28).

So when Elizabeth meets the new mother, Mary, she exclaims with a loud voice:

Luke 1:41-43 DRB

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?

Parenthetically, this latter phrase 'Whence is this to me,' or 'Why am I so favored that...,' is an act of high veneration or respect or gratefulness for a privilege which one feels unworthy to have been blessed with: beyond that of a simple relative coming to visit! It echoes precisely the kind of language used in Ruth:

Ruth 2:10 DRB

She [Ruth] fell on her face and worshipping upon the ground,* said to him [Booz]: Whence cometh this to me, that I should find grace before thy eyes, and that thou shouldst vouchsafe to take notice of me, a woman of another country?

^{* There was a sense in the ancient world where 'worship' or keen veneration or reverence was commonplace in the slave-to-master relationship, and not an act of adoration as such, as to God.}

The Implications

In other words, if you cannot, with a happy smile, declare that Mary is the Mother of God, you are a heretic who worships two Christs. One who, when Mary was bearing Him or teaching Him His first words, was only human. But later became God at some later stage. Or who is human without being God in some way.

Or, if you say Christ is not God, unless He is also the Father and the Holy Ghost, ('But Jesus wasn't the whole Trinity') you are a heretic also. To be God, is not 'to be the Trinity.' It is to be of that one Essence which the Trinity shares: God. Otherwise these people could not affirm Jesus is both Son of God and God, equally: Son meaning 'of the same substance as the Father'. Not being an addon for the Father, in which case He would be ontologically separate from God.

In other words 'God is Jesus' and 'Jesus is God' are semantically not equivalent.

If Mary became His mother and bore Jesus Christ, who is and at every point was God incarnate, which Scripture affirms, then she is, inescapably Mother of God.

A Few Objections

Objection 1 — Is Joseph not also called the 'father' of Jesus?

Joseph is called the father of Jesus, but there is at the same time two Scriptural facts and qualifications which help us learn that Joseph is not a real father at all, but a foster father: before there could be any martial union (necessary for Joseph to be a father in any sense above a protector), Jesus was conceived by Mary (Matthew 1:18) (at which time she became the mother of Jesus, since she possessed at that time the only thing in which motherhood must needs consist).

In fact, St. Luke parenthetically notes that Joseph was only “supposedly” to be the father of Jesus (Luke 3:23) (i.e. mistakenly). Not that he was in fact. Consider how he qualifies:

The same Greek word (νομίζω nomizó—I consider, suppose, think, deem) used by Luke in his Gospel to say that people ‘[mistakenly] thought’ that Joseph was the father of Jesus was used also by him in Acts, in quoting Paul saying that Simon “thought” or ‘assumed’ he could buy the gift of the Spirit, since known as the sin of Simony (Acts 8:20); obviously, a grievous error.

Not surprisingly, Mary is never said to be anything but His mother: no 'His mother, as was supposed.'

Objection 2 — God has no parents, since He is said to have 'neither mother nor father'

Mary was the mother of a Person who happened to be at all times God. She did not give birth to God as in be His origin: she accepted the request that she become a mother, namely of 'the Son of the Most High', which she thought, presumably, was pretty cool, and replied humbly: "Behold the handmaid of the Lord; be it done unto me according to thy word."