Hm. Speaking as somebody who's a bit of a noob in her own right, and just dragged my Mom back into gaming, I'd say step one might be to buy a used 3DS for her.[footnote]Often if you don't mind gambling on whether a thing works simply because it's been turned in without power cord and nobody knows, shopgoodwill.com is a good place to look. It's largely confused volunteers not knowing much about how these things cost. I got my mom a 3ds in perfect shape, with case, for around 50 dollars - I just had to take that chance and have some faith that it was a sturdy little workhorse and would charge up just fine.[/footnote]
The 3DS has a lot of games that I think are great introductions to gaming. They might seem a bit 'kiddie' or 'non-hardcore' but you may have to work your way up to things that are more complicated. Try Animal Crossing and Pokemon X/Y to start out with. AC is pretty easy and very relaxing, and Pokemon X/Y is one of those games that is very forgiving but actually does have some complexity you can sit and think about.
Your girlfriend may be a little hesitant because she's looked at what you play and gone "oh jeez that's fuckin scary", so these games will kinda show her the lighter, fluffier side of things as well, lol.
When she's hooked via the 3DS, think about PC games. If you can find a server to play on together, Minecraft is a great game, especially if she says something along the lines of "I love Animal Crossing, I just wish I could build whatever I wanted". That'll introduce her a little more towards survival and combat, and most of all, it's something you can do together. You might want to go ahead of her just a tad and build up a little bit - not a whole huge fortress, but enough that as soon as she hits minecraft, you can hand her some stone tools (instead of useless wooden ones) and you can huddle together in a little hut while avoiding all the zombies outside when night comes hahaha
Also look at games you can play cooperatively with her, too. If she wants to try FPSeyness, what about Warframe? It's not that complicated, it's extremely satisfying "I'm a motherfucking ninja in space flippystabbyshootyWAHEY" gameplay, it's free to play, and it's something where you guys can competitively play together. Soloing is totally possible in the game so she isn't beholden to you being there, and she can always dip her toes into matchmaking too. Also it's free-to-play so, you know, not like you're losing any money or anything.
The other game you could play cooperatively with her might be a MOBA like LoL, but that might involve bringing in another person or setting up custom games where it's you + a bot vs. all bots. This is going off my experience playing with people I like in LoL, I just plain don't like matchmaking because I know I'm a noob. I'm also a noob with anxiety disorders but I digress
Most importantly, though... ask her what she wants to play!
Maybe something's already caught her eye. Be there to give help when she asks for it, but don't spoil it for her (as hard as this totally is in some games). Make sure she knows the controls, give her access to your gaming stuff if she wants to try a game on your ps3 or something, and hey, take the advantage to totally sprawl out on the floor behind her and be her living armchair and get some good cuddles in while you enjoy watching her discover and enjoy what you like too!
