The first one that comes to mind is Oz Fox. His real name is Richard Martinez, I think. Oz Fox is his stage name.
Cliffy Huntington's (The Huntingtons, Main Line Riders) real name is Cliff Powell. In fact, all four guys in the Huntingtons used only their first name and took the surname Huntington (much like the Ramones).
Mike Naeyert vocalist of Barren Cross and Barebones goes by the stage name Mike Lee and Mike Drive and Michael D.R.I.V.E.
Bob Dylan's birth name was Robert Zimmerman but he legally changed it to Bob Dylan.