She's about to turn 7 as I write this and tabletop games are nothing new to her having been surrounded by my hobby for all of her life and of course she was part of what led to the end of the 1st Chapter.
To help in my attempts to introduce her to games I have over the past few years bought some games appropriate for younger gamers.
Catan Junior - The parent version of this game is seen by many people as the gateway game for the board game hobby. I don't necessarily share that opinion but I do see why it has that reputation. That aside the game is a great introduction to the hobby as a whole and I plan to play it with my wife and daughter soon.
Hey That's My Fish - It's a game about penguins trying to catch as many fish as they can before the ice falls apart below their feet. We've played this a few times, actually played might be too strong a term. The structured play (albeit limited) is something that my daughter hadn't quite grasped when we played it previously. I plan to re-try the game now that she's a bit older (she was 5 when we tried it before so definitely a bit young).
Straw - I've mentioned this in another post and the core of the game is arithmetic. Something my daughter has a very strong grasp of. It's one of those games that is quick to play and is very simple mechanically so an excellent candidate to use in a family environment.
Alongside these there are a plethora of other games that can work in the family environment not least of all are roleplaying games. Rather than me listing a whole bunch of them here I'll link to a post by WJ Walton of The Escapist. For now though I think I'll stick to board games with my daughter and see how that goes before trying out some of the RPGs listed.