We call Mary the “Mother of God” because that’s exactly what she is. To understand why this is true let’s back up a step and consider who and what her son, Jesus Christ, is.
Since his Incarnation Jesus has had two natures, divine and human. These natures are completely united (meaning he is completely God and completely human). The technical word for this is the hypostatic union.
Although Jesus has two natures, he is only one Person–God–the Second Person of the Trinity. For this reason Jesus is properly called the “God-Man.” (By the term nature we mean what Jesus is; by the term Person we mean who he is.)
Since the Son born to Mary is a single Person (and that Person is God) with two natures, Mary can rightly be called the Mother of God. The Person she’s the mother of is God–he’s not, strictly and philosophically speaking, a human person, as we are.
This doesn’t mean that Mary existed before God (which is an impossibility). She is a human person with a human nature. She existed before Jesus’ human nature was created.
Here’s another way to look at it. A woman can’t be the mother merely of a nature. She can only be the mother of a person who possesses a nature. When a child is conceived he is a person, not just a nature.
The same is true of Jesus, or else he would be two persons in addition to having two natures. Since Mary is the mother not of Jesus’ human nature but of Jesus the God-Man, a divine Person, she is rightly called the Mother of God.