My then-girlfriend, now-wife, and I had a ton of fun with Braid (we approached it and Goo similar to your taking-turns model). She's a mathematical puzzle-solver type though, so your mileage may vary.

We're supposed to be doing Portal 2 together at the moment, but that hasn't really coalesced due to ongoing grad school for both of us, so I can't comment on its value. The original worked great in the same system, though -- although she tended to figure out how it was supposed to work and I tended to do the actual execution on that one, so it depends on your respective skillsets and how satisfying you find that.

My brother and his girl both enjoyed World of Goo, I should ask him what else they've done.

I've played the following games with my wife, most of them have already been mentionned here but I can confirm they lead to great coop experience :

Portal 2 (really awesome in coop. You'll have to buy 2 copies though, and have to use Steam. You'd better wait for some Steam sale, maybe christmas ?)
Magicka (same as Portal 2, steam-only game). Really hilarious, though a bit unforgiving at times.
Lego Star Wars 1&2, Lego Indiana Jones
Titan Quest (any hack'n slash will do, really. But this one is good, there not too much blood, and the ancien Greece fantasy theme can be appealing for girls)
Trine, and soon Trine 2
the Heroes of Might and Magic series, if you're into this type of games. Heroes V is a good starting point, Heroes 3 is generally considered better but it's more hardcore.

Maybe also building games such as Minecraft or Garry's Mod, but I haven't tried those yet.

Try coop of indigo prophecy. :) she plays as the chick you do the guys.

I am never get tired to advert Walkover...

Co-Op Alienfighting Fun for free and me as part of the developer team :)

www.walkover.org

Ooooh this will be good, especially the asylum level ...

Oh, oh, what about Amnesia? You guys can take turns to scare yourselves silly. Let her play the Storage level!

I mentioned her the game, and this:

http://www.reddit.com/r/gaming/comments/m233o/strip_amnesia_i_think_im_winning/

She said like: No way! :D

eh. you did it wrong. first you have to open a bottle of tequila and then you suggest the amnesia coop.

Minecraft is quite fun together, Portal 2 co-op is pretty cool, and I'm also hoping the upcoming Torchlight 2 will be good :)
In addition we've had fun with infinity engine games. But overall it's kind of hard to find nice non-violent/twitchy co-op games.

I'll throw in another plug for Puzzle Pirates. It's MMO, built around casual cooperative gameplay using puzzle performance to make ships move and win swordfights. One of the original free-to-play paid games, without being pay-to-win. Big timesink if you get into it.

Portal 2 is out of the question... don't really have 23 EUR right now to buy one, let alone two copies. Maybe we'll play Portal, the original, if she likes it.

And co-op in Trine looks like it needs a controller, but as a PC gamer me no has controllerz. Maybe I'll find one for the weekend, though.

Checked out Trauma. That's a game she'll definitely like (and already have it through Humble Bundle).

My last girlfriend and I played some Trine together, but she wasn't really that into it. The only thing I ever really got her into was playing point and click adventures with me where we would work together to solve puzzles, but even then she rarely wanted to play. Some people just aren't gamers and there's nothing you can do about it.

My best friend and I recently played Magicka on the same PC and had a blast with it. We also played Kane and Lynch on the same PC but I am guessing your girl would frown at that. Kane and Lynch 2 removed same-PC coop, which was BS.