Myself: Bob. That's what I always name my character in Pokemon games. Or pretty much any games, for that matter.
Rival: Hugh. Because it's the default, and when I say that "Hugh blah blah blah," people should immediately know who I mean.
Pokemon: Other than solo runs or nuzlockes or the like, I don't nickname. I go through far too many pokemon - catching and training and evolving and trying out movesets and putting them on my team and taking them off my team - to come up with anywhere near enough good nicknames.