In Gold I named the Rival Knuckles (after the guy from sonic, he looked like him.)
In Heart Gold I named him AxlRose (no space) the real Axl Rose is a red-headed douche, and so is your rival, so it fit.
Well, to be honest, I used to use default names until he just ticked me off, so now my rival is always named douche as a reminder why you never have a rival in a pokemon game, because they're douche bag's.
Though it does not make any sense, on my first run of Platinum I called my rival Pickle... I don't know why I just did, it still makes me snicker. Also I named him the good old fashioned Fart so your rivals mother would say "My Fart took off like a rocket as usual today."