Some do, some don't. Gabriel, for instance, did choose a new name at chrismation. I'll let him tell the story if he wants. For me, I was given a saint's name at birth, with the full intention of my parents that I should be named after a saint--which is an Orthodox practice, despite their being Protestant. I felt it wouldn't be right for me to forsake my given name, which is really an Orthodox name already.
At any rate, whatever name you choose should be done in consultation with your priest, godparents, and biological parents--after all, they did name you in the first place.