In the older Pokemon games, excluding the Hoenn games, and first Unova games, you could give your rival any name you wanted, pretty much. You could name your rival in the original games, their remakes, G/S/C and their remakes, D/P/PT, and the B2/W2 games. It has since then been absent from every Pokemon game. Naming your rival is no longer a function in any Pokemon game.
What did you name Blue, Silver, Barry and Hugh? What did you think of being able to give your rival whatever name? I always found it fun. By the way, I name my rival "Kir" when I play the older Pokemon games.
I usually use the rivals’ regular names. I used to name Gary and Silver nasty names as a kid, but that’s not something I do now.
Sometimes, when I name my trainer after a character from a game (ex. Robin), I name the rival (if not friendly) after their nemesis (ex. Grima). If friendly, I name the rival after an allied character (ex. Chrom).
“You’ll go no further! For her sake, I will not fail!” -Ash, “Phantom Brave