"Players are able to choose a different name for Blue (RBY/FRLG), Silver (GSC/HGSS), Barry (DPPt), Hugh (B2W2), and Trace (LGPE). In the Hoenn- and Kalos-based games, the player character not chosen will serve as a rival."
Anyway, I can't really remember my rival names back in the original two generations. I'd either pick the default Gary or whatever provided for the rival. I also likely defaulted to Barry in generation IV because Barry looked like a Barry to me.
For more recent playthroughs, when my trainer name was a reference to a video game, I would typically use an antagonistic name for my rival. Some examples included:
White 2: My trainer name was Lilith and my rival's name was Jack (Borderlands series)
Black 2: My trainer name was Aloy and my rival's name was Sylens (Horizon Zero Dawn series)
Let's Go Eevee: My trainer name was Aurelia and my rival's name was Alistar (Borderlands series, though I was technically playing as the antagonist).