As others have suggested, co-op games are certainly the way to make things interesting and fun, especially when there's going to be an obvious skill imbalance. Also try to pick things with a very shallow learning curve - if she hasn't played games before, just getting the coordination with a controller or mouse can be frustrating enough.
Games that have a low punishment for failure are going to be key when someone is first starting. This isn't quite the same as a shallow curve, but you want a game that is forgiving of your errors whilst you learn to play. Similarly something that isn't high pressure is probably good for early games. Left 4 dead, despite its excellence, probably isn't the best way to get into things (but will make an incredible game later if she gets into it!)
There are a couple of games I've found that can be really great fun in this way, and depending on /her/ tastes, you should find something:
First there's Trine (and its sequel). You can pick this up quite cheaply, and it's a lovely fairytale of a game, beautifully drawn, gently but excellently narrated. It's a 1-3 player co-op platformer/puzzler (I played it with my partner who loved it) and having more people massively increases the fun. It also doesn't do the usual gendered thing with games of having "chick-armor" or "all people are male" - the female character (one of the three players) is very nicely done. Set on an easy mode it's simple to learn, works excellently with controllers and doesn't require too much coordination for a newcomer. If you pick your timing right, you can get it for about five dollars on Steam.
Another great coop game is Civilization V. I know it's not the most hardcore in terms of strategy of the series, (and I'm presuming as a gamer you know the series) but its very easy to learn and playing as a team, either hot-seat or with two computers, is very satisfying. A more experienced player can provide cover the learner in terms of military protection etc, or just set the game on sufficiently easy mode whilst she learns the basics. In coop mode you can learn a section of the game at a time whilst your partner takes care of the rest, so she can focus on military strategy and world domination whilst you build the empire to fund it, or she can learn to manage cities to produce culture and science whilst you cover her borders. The turn based nature of the game makes it easy for teaching someone how to play, and it offers a ton of depth and replayability.
On the RPG front, Torchlight is marvelous, with its sequel being a great 2-player game. It's diablo-esque, but maintains the joys of D2. It can get a bit hectic on occasion, which is very frustrating, but with a co-op game again you can cover her.
On the FPS/strategy, Orcs Must Die (2) is a nice one, but does suffer horrendously from a couple of things - it has a nice learning curve, but can get overwhelming fast which leads to frustration. Also it has tongue-in-cheek cliched characters which at first will look rather like the female is supposed to be the stuff of adolescent fantasy. It's not as bad as many out there, but let's just say that her armor is less than optimal in some regions.
Hopefully that should be something to get you going. Ignore the people who say "Don't do it" - of course you should try out new hobbies together, and you may find an excellent way to have fun together. My partner and I game together often, and sometimes at long distance is a great way to spend time "together" when you're apart.