I used to always name him after my best mate, when he still played the games. He stopped back at the R/S/E gen and since then it always depended on what I felt like when I started a game, bear in mind I'd start a new game every three or four months.
Currently, however, I have a theme. My character's name is Django on both my SS and Pt game. On each game, nicknames follow a rule. On SS they are named after actors/comedians/writers/musicians I'm particularly fond of. On Pt they are named after fictional characters, often ones created by or portrayed by ones I've named on my SS game.
As such, my rivals are named appropriately. I'm named after Django Reinhardt but when one thinks of Django nowadays they go straight to the recent film and rightly so. On SS my rival is named Caprio (DiCaprio is one character too long), reflecting his antagonistic role in the film. In Pt he is named King, reflecting the eccentric camaraderie.
It occurred to me after choosing the rival name on SS it fitted appropriately with his parentage.