Jesus is the earthly name of the Son of God. The angel spoke to Joseph concerning Mary, "And she shall bring forth a son, and thou shalt call his name JESUS: for he shall save his people from their sins" (Matthew 1:21). He was not called Jesus before His birth because Jesus refers to His humanity.
Isaiah 9:6 begins, "For unto us a child is born, unto us a son is given." Notice that the son is given. That which is given already exists. As the Son of God, Christ existed into eternity past. He was rich but became poor so that we through His poverty might be rich (2 Corinthians 8:9). But notice also in Isaiah 9:6 that the child was born. Birth indicates the beginning of something. The Word, the Son of God, had never been a child before He was given to this world and born as a baby in the manger in Bethlehem.
When Jesus was born, He did not subtract anything. He had always been God and was still fully God. However, He did add something. Now, we was also man. The Word, who was God and who created all things (John 1:1-3), "was made flesh, and dwelt among us" (John 1:14). He always has been and always will be God, but He became a man.