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Had no other choice but to name him Richie (the kid that had a Pikachu with hair in the 1st Pokemon series), I'm unsure if I used that name on my Diamond's rival though.
I always name mine Chris, mostly cause my boyfriend and I are very competitive and I find him to be my rival in many ways. Haha, so I've always used his name. Before we were together though, I usually went with one of the defaults.
Almost all the time my rival's name is Gary, simply because he was the original rival of Ash, and my character's name is always named Ash, boy or girl, Ash fits either.