It wasn't through Mary's biological make up which gave flesh to Jesus Christ. Mary was overshadowed by the Holy Ghost...therefore Mary's biological make up had nothing to do the Incarnation. It was ALL God's work. DHK is correct in the use of the word "surrogate" in that Mary did carry the child Jesus.
God is eternal. God has no mother. Jesus Christ is the Eternal Son of God. While Mary was the mother of Jesus as a man; the eternal Son of God had no beginning (Micah 5:2; John 1:1). Mary cannot therefore be called the mother of God, and is never so called by Bible writers. Yes, Mary was blessed above other women to have had the privilege of bringing the Savior into the world, but this allows no foundation for calling Mary the Mother of God, Queen of Heaven, Mediatrix, etc. The only mention of Queen of Heaven in the Bible refers, not to Mary, but to a heathen idol (Jeremiah 44:18-19).
Isaiah 9:6 For unto us a child is born, unto us a son is given: and the government shall be upon his shoulder: and his name shall be called Wonderful, Counsellor, The mighty God, The everlasting Father, The Prince of Peace.
Isaiah 9:7 Of the increase of his government and peace there shall be no end, upon the throne of David, and upon his kingdom, to order it, and to establish it with judgment and with justice from henceforth even for ever. The zeal of the LORD of hosts will perform this.
Note: It is the child who is born, not God---and it is the son who is given. The "son" is "given" by God (John 3:16). All the attributes which are given to the "son" are attributes of God, Who is eternal.