In R/B/Y, I always named my Rival "I" for some odd reason, and that name is strangely nostalgic now...
In G/S/C, I named my Rival "Zaphod", and that has recently been done for HG/SS.
In Emerald, I never got to name my rival.
In Platinum, I named my Rival "Steef" after a good friend of mine...
I have no idea what I'll name this one, it will depend what he acts like during the first meet up with him. (I don't know, and would rather not know until I actually play the game) If he is nice, I'll probably name him after a friend, if he is harsh to some degree, I'll probably come up with a harsher sounding name.
I won't name him like 4chan names things though. (I know a Bidoof named "Handjob" for instance)