I've already attempted, somewhat, to prove the Theotokos here. Quotes are in bold and orange, my defense is in regular print.
— According to Catholic doctrine, because she is the mother of Jesus, and Jesus is God, therefore Mary is the Mother of God.
According to Catholic doctrine, and according to simple logic (If A is the mother of B, and B=C, then A is the mother of C). This is called logical equivalence, by-the-way.
_____The Incarnation means that Jesus was both fully God and fully man. Mary was only the mother of Jesus as man, and not the mother of Jesus as God. According to the Bible, the world was created through Jesus. The Bible says,
There are SO many problems with this. Tell this to any mother. I dare you. Because essentially what you are saying is that someone is the mother of their child's nature, but not of the child's personhood. That's just wrong. A mother gives birth to a person, not a nature. Mary gave birth, and is the mother of, Jesus as a person. And who is Jesus as a person? He is fully G-d and fully man. It is ridiculous to separate Jesus' two natures, and to say that a mother gives birth only to natures and not persons.
Jesus cannot be separated as human and divine. He is both distinctly human and distinctly divine, fully and completely. But it is all in His personhood. Following the logic just given by our Fundamentalist amicus, one could say that when Christ died, it was just His human nature: that's heresy! It's called Nestorianism, and ironically it led to the First Council of Ephesus in 431 which proclaimed Mary as Theotokos in order to better illustrate that Christ was fully human and fully divine. What our Fundamentalist hoaloha is arguing is anti-Christian and a heresy.
_________"God, who at sundry times and in divers manners spake in time past unto the fathers by the prophets, hath in these last days spoken unto us by his Son, whom he hath appointed heir of all things, by whom also he made the worlds." (Hebrews 1:1-2, emphasis added)
_________"For by him [Jesus] were all things created, that are in heaven, and that are in earth, visible and invisible, whether they be thrones, or dominions, or principalities, or powers; all things [including Mary] were created by him, and for him: And he is before all things [including Mary], and by him all things consist." (Colossians 1:16-17, emphasis added)
_________Jesus said unto them," Verily, verily, I say unto you, before Abraham was, I am." (John 8:58)
_____Jesus existed before Abraham was born. That means that He also existed before Mary was born. Jesus said,
Ah, here we go. It's a legitimate question, I suppose, on the surface of it all. It would certainly be a great argument against any normal person (how could a 54-year-old be the son of a 35-year-old). But when one starts to think about the Trinity, it becomes evident that this entire argument is totally stupid.
Let's look at what is being argued closely. Our Fundamentalist neighbor is saying that Christ existed before Mary (before everything!), and therefore could not come afterwards. But this argument is a better argument against the fact that Christ was born than it is that Mary is His mother. Jesus existed before Mary, but entered into time and became flesh. Jesus, as G-d, came before Mary, but entered into time and became flesh. So what that He came before? No one is arguing this. Mary isn't the mother of G-d the Father, Who did not take on flesh; nor is she the mother of the Holy Spirit, Who (as the name suggests) did not take on flesh.
I don't think this person even realizes what s(he) is arguing! What they're arguing is that Jesus was not born to Mary, not that Mary isn't the mother of G-d.
_________"And now O Father, glorify thou me with thine own self with the glory which I had with thee before the world was." (John 17:5, emphasis added)
Again, this person isn't understanding their own argument. Maybe I'm just missing it (if so, PLEASE someone let me know!)
_____So Jesus existed even before the world began. Jesus came first—not Mary.
The Catholic Church does not, in anyway, believe that Mary came first. She came after G-d (much after). However, G-d decided to become flesh and dwell among us, and He chose Mary to be His vessel. She was the mother of G-d, not G-d the Father or G-d the Holy Spirit, but G-d the Son.
This is why it is more appropriate to call Mary the Theotokos, or God-bearer. It is much easier to make sense of Mary bearing G-d in her womb than it is to say that Mary is the mother of G-d.
Part 1, Part 2, Part 3, Part 4