The Fuji F10 is a great little camera, as recommended above. It has a good sensor inside it that makes it ideal for taking pictures in lower light conditions, like inside restaurants and houses. The sorts of places your average 13-yr old may not think twice about taking a photo, only to usually find it comes out dark.
It's a little bulkier than some other models though; if you want something a bit thinner, have a look at the casio range. The Casio S500 is about your price range, and is a great camera. Maybe not as well built or tough as other cameras though, due to its size.
Finally, you can't go wrong with any Canon compacts. They're always well-made and give good results.
Also, I'd strongly suggest you either get a case or at least a screen protector. I use these on my camera and phone, and they're every bit as good as their videos suggest:
http://www.shieldzone.com/