I have no idea what a Wonder Trade even is, but I remember being shocked when I booted up Diamond and was asked to name my rival, as I had only played Emerald before then. I was like "who is this guy, Professor Birch wasn't dumb enough to not know May's name!". I think I definitely called him Sam for some reason, and Barry a few times.
Buying LeafGreen I think I called the rival Gary because of the anime. In HeartGold I called Silver "Murtaugh" after the Eragon character because it felt somehow fitting (also because I planned to have a Dragonite "Glaedr" to go with my traded in Salamence "Saphira" - owner "Eragon").
Once I started reading Pokémon Adventures however, I've generally started naming both my player and the rivals after their manga counterparts, so Barry = Pearl and Silver = Silver. Kinda jars when you can't rename Dawn or May (Wally stays the same in the manga) or the rivals in Gen V - so I tend to be more likely to name my character in Black or Emerald after myself, or some other name like the aforementioned "Eragon".
In Black 2 I believe i just stuck with the name Hugh as I had kinda drifted away from pokémon somewhat for a couple of years after Black's release.