As the parent of a 7 year old girl, and as someone who has a reasonable stable of iDevices as well as a Nexus 7, Kindle Fire HD, and Kindle Fire (edit: all of which the 7 year old has played with), I'd say it depends significantly on how much you want to invest and what the usage will be like.
I think the iPad mini is ridiculously overpriced compared to the competition, but in terms of games, it has a much better variety, and it likely always will. Games will fit onto 16GB, no problem. And you'll never think the iPad mini is low-quality.
On the other hand, for movies and shows (which, despite the initial desire to put learning games on it, will creep onto any device), the ability to get 32GB on a Nexus 7 for $250 along with the trivially easy rooting-and-using-a-USB-OTG-cable-to-add-infinite-storage-capability option, it's hard to argue against the 7. Also the GPS and offline maps might amuse her during trips.
So, from a media standpoint, I think the Nexus 7 wins. There are also most of the popular kid's games on it as well, and the percentage of kid's games on it is growing.
The Kindle Fire options are both decent, but they have fewer games than the iPad mini or the Nexus 7. The Fire HD does have vastly better sound than the other options, though. If she'll watch a lot of movies, it's worth considering.
The Kingston Wi Drive will add 32GB of media wirelessly to any of the devices for $70, so keep that in mind as an option as well.
If it were me, I'd spend $220 on a 16GB Nexus 7 and cheap case (kids drop things), but I couldn't fault you for any other option. They all have plusses and minuses, and none of them are truly bad options for kids, and all of them are kid-friendly.
Note that the Nexus 7 will soon support multiple users, and the Fires already do, so that's a selling point until iOS adds the capability.