I name my character Kain for my first file of SoulSilver as a reference to Final Fantasy IV and I named my rival after one of the villains from that game. My friend got HeartGold on the same day and named himself Cecil (also a FF IV reference) and named his rival after another villain from that game.
In every other file of these games I've named my character THIEF as a reference to Link's Awakening. One of the best Zelda games of all time. It's great when people say "THIEF! You are a very kind trainer!" or "THIEF? What a wonderful name!"
"You think our rage a weakness? Then let me show you... how wrong you are!" - Edge, Final Fantasy IV
I named the female main after "Karen Thorn," a character I've been drawing for.. over a decade now, hard to believe. They both have huge, bright eyes, big, happy smiles, and volumes of hair. Not to mention that, to paraphrase one online cartoon, they're both just so bouncy and round! ^_^ (In one way or another)
Now I just have to draw some pictures of Karen dressed as Rosa in certain Pokémon-esque situations (exploring Join Avenue, hanging out at her Dream Island, making a film at Pokéstar Studios, etc.).
(Also I really didn't like Nate's appearance, and since I had already played as Hilbert in Black, using the name "Amesang," I decided to switch it up a bit.)
EDIT: Ynni's post reminded me that I had named the rival, Hugh, "Grey," as a reference to the games titles', as a pseudo-anagram of "Gary," the original rival, and because of "Fifty Shades of Grey" since I had originally (and still do, occasionally) drawn Karen as a succubus.
While I'm editing I'll mention that I named my Black character "Amesang" because it's the last name for a different character I draw (female as opposed to male, however), as well as being my account name on YouTube and my Gamertag on Xbox LIVE. Had it not already been taken by other people on Twitter and Facebook, I'd have used it there, too (assuming I bother to ever sign up).
"Help! Help! They've destroyed my cousin's brain!! Oh, my god! They've already milked you, haven't they?!!" -- Phoncible P. Bone