Woah, that's a hard one. If she's not a gamer-chick to begin with then you'll have to be careful "popping her cherry" as it were. I dunno what to suggest, it seems to me that whatever game you get into is going to have to appeal to her interests. I'd wonder what sort of shit she likes do IRL etc?
Non-Geek can imply so many things.. she a jock? Academic? Womens-Lib kind of girl?
Ultima Online is pretty gay, but it's free and lots of people can still be found playing on private servers... err... shards. She into horror type stuff? If so there's a plethora of console RPG options.
Man, chicks confuse me. That, or I scare 'em all off.