Is there any tradition that informs us of how old Mary was when she gave birth to Jesus?

Despite people believing that she could have not been no older then 14 is not true. Mary like most Jewish girls was in her mid teen years or even late teen years (Source: New King James Version, Study Bible, Cultural Backgrounds, @google books)

There are other several sources that claim that Jewish girls married early-mid teen years, but it seem that some scholars only think Mary was no older then 14, despite the fact that Jewish girls did get married over 14 years and probably most likely did. There's no proof that 12 was the customary age of marriage
{Source: The Reign of God is Such as These: A Socio-Literary Analysis of Daughters in the Gospel of Mark; Sharon Betsworth; @googlebooks}

Matter of fact the idea of most people dropping dead around 20-35 is also a myth.

"the average life expectancy for males in the U.S. is 76.3 years, and for females it’s 81.2 years. By 2030 these numbers are projected to rise to 79.5 and 83.3 respectively.

But, again, these are life expectancy numbers, which are calculated based on how long someone is likely to live at birth. It does not give an accurate picture of what age the average adult in a society is likely to reach. That number has actually remained fairly steady throughout history."

The Magificat or the Canticle of Mary (Luke 1:46-56) mirrors Hannah's prayer 1 Samuel 2:1-10 so closely that one would have to think that Mary found her words in through reflecting on scripture.

And Hannah prayed:                             And Mary said:
My heart exults in the LORD,                   “My soul proclaims the greatness of the Lord; 
my horn is exalted by my God.                  my spirit rejoices in God my savior  
                                               For he has looked upon his handmaid’s lowliness;
                                               behold, from now on will all ages call me blessed.  
                                               The Mighty One has done great things for me,
                                               and holy is his name
                                               His mercy is from age to age
                                               to those who fear him.
I have swallowed up my enemies;
I rejoice in your victory.
There is no Holy One like the LORD;
there is no Rock like our God.
Speak boastfully no longer,
Do not let arrogance issue from your mouths.
For an all-knowing God is the LORD,
a God who weighs actions.
“The bows of the mighty are broken,            He has shown might with his arm,               
while the tottering gird on strength.          dispersed the arrogant of mind and heart.
The well-fed hire themselves out for bread,    He has thrown down the rulers from their thrones
                                               but lifted up the lowly
while the hungry no longer have to toil.       The hungry he has filled with good things;
                                               the rich he has sent away empty 
The barren wife bears seven sons,
while the mother of many languishes.
“The LORD puts to death and gives life,
casts down to Sheol and brings up again.
The LORD makes poor and makes rich,
humbles, and also exalts.
He raises the needy from the dust;
from the ash heap lifts up the poor,
To seat them with nobles
and make a glorious throne their heritage.
“For the pillars of the earth are the LORD’s,
and he has set the world upon them.
He guards the footsteps of his faithful ones,  He has helped Israel his servant,
                                               remembering his mercy,
but the wicked shall perish in the darkness;
for not by strength does one prevail.
The LORD’s foes shall be shattered;
the Most High in heaven thunders;
the LORD judges the ends of the earth.
May he give strength to his king,
and exalt the horn of his anointed!”
                                                according to his promise to our fathers,
                                                to Abraham and to his descendants forever.”

That scripture, which would have been burned in to her Immaculate Heart, by means of a singular grace would have certainly been with her at a young age when she would be betrothed. One tradition (not Sacred Tradition), which is mostly forgotten by popular imagery, is in that Catholic Encyclopedia article Drew linked to which says that Mary was a consecrated virgin raised in the temple, that her parents Anne and Joachim prayed for her and when they got her, gave her up to the temple where she would have been able to absorb scripture like Anna (Luke 2:36-38), who would have certainly known all about her if that were the case.

Some understand that Mary was about 15 years of age based on the juxtaposition of Herod's Temple and the real Temple who is Christ Jesus.

Herod began building what would be called Herod's Temple in 20 BCE in the 18th year of his reign. Josephus says it was the temple of God.

1. AND now Herod, in the eighteenth year of his reign, and after the acts already mentioned, undertook a very great work, that is, to build of himself the temple of God, Ant. XV. 11.

The work began. So too Mary would be born and raised up over those 15 years to 5 BCE when the temple was finished and the virgin conceived. 9 months later the real temple of God was born in whom we have redemption through His blood.

PS. Manuscripts of the Protevangelium of James have her age ranging from fourteen to seventeen.

And she was sixteen1589 years old when these mysteries happened.

1589 Six mss. have sixteen; one, fourteen; two, fifteen; and one, seventeen.

In Mark 5:41-42, a 12 year old is referred to as a little girl. Mary was always called a woman in the Bible. So there is a strong possibility that she was much older than 12. Likewise, in Luke 1:42, Mary is referred to as a woman - "Blessed are you among women" - even before Jesus is born.

I have more evidence to support the claim that Mary was older than 20 when she gave birth:

God in the Bible called all those below the age of 20 as children

Numbers 32:11 “Surely none of the men who came up from Egypt, from twenty years old and above, shall see the land of which I swore to Abraham, Isaac, and Jacob, because they have not wholly followed Me.”

Deuteronomy 1:39 “Moreover your little ones and your children, who you say will be victims, who today have no knowledge of good and evil, they shall go in there; to them I will give it, and they shall possess it.”

Mary was always called a woman in the Bible. In Luke 1:42, Mary is referred to as a woman - "Blessed are you among women" - even before Jesus is born. This proves that Mary was older than 20 when she gave birth.

Proof from Bible manuscripts that Rebecca was 20 when she married Issac.

http://www.jewishencyclopedia.com/articles/12610-rebekah According to the Book of Jubilees (xxxi. 8-11, 48), Jacob, when he arrived home, found his mother alive; and she afterward accompanied him to Beth-el to accomplish his vow (comp. Gen. xxviii. 19-20). She died at the age of 155, five years before Isaac's death (Jubilees, xxxv. 1, 41), this determining that her age when she married was twenty years.

The Book of Jubilees is found in the manuscripts of the Dead Sea scrolls and is part of the canonical Bible of the Ethiopian Orthodox Church.