I just bought a tablet for exactly that purpose, but for a 2-year-old. In the end, I bought the second-cheapest 7" Android tablet I could find (yet still was running 4.0) because of two things:
a. If it breaks, we just broke a cheap tablet and not an expensive one.
b. It's for running a few games, not much more. Maybe hangouts if MrsBarn uses it when travelling.

In the end I got an off-brand Prestigio PMP3570C which is worth every cent of the roughly USD$100 I paid for it. First reactions were:
1. The screen resolution and sensitivity are definitely not Nexus 7 standard and a 2-year-old can tell.
2. It is noticeably slower, but not unuseable. Just... slower.
3. It is much more fragile than a more expensive tablet. This is NOT Gorilla Glass. GET A CASE. (and, of course, it got dropped and cracked. waiting on repair now).
4. The battery life and brightness are fine for indoor/aircraft. Good enough. Charges via included charger OR microUSB.
5. It's got a microSD slot, easily accessible, very nice.
6. Speakers aren't great and while it can play HD content, it struggles and isn't watchable when streaming over DLNA, unlike the Nexus.

If this is to be a tablet for a kid who will use it heavily, the screen thing may be an issue. But if this is a tablet for a kid who has never used one before and you will be in the position to upgrade it soon, then I'd get a cheap one.

As for size: I really like the 7" factor and find that for small hands it's just easier to hold. It's also easy to throw in a pocket or a diaper bag (or child's backpack) and take with you. If I had a 5"+ phone then maybe I'd go for the 10" but right now the 7" comes with me almost everywhere.

As for games: I also don't know the child in question, but I've found that the most popular game we've got is Puzzingo. It may be a bit young (keep in mind I'm basing this off the recommendation of a 2-year-old) but it's cool nonetheless. Few free maps, buy loads of others for not that much. I like it and would easily recommend it. All the other kids' games I have, though, are certainly too young. Mostly alphabet/numbers games.

But for not-necessarily-kids' games, Little Things Forever is sort of a "where's waldo" scavenger hunt that can get pretty intense some times. Might be a bit old, but you could play it together.

I'm just waiting for the day that I can get my kid hooked on Carcassonne or Ticket to Ride.

On the other hand, if your daughter likes My Little Pony, the Android game is of VERY high quality. Right down to how they got all the voice actors back to do loads of in-game dialogue. Just make sure to restrict down the rights of the game so it's not constantly annoying you with notifications and online connections. But overall I'd recommend it and that's probably saying far too much.