Ive recently bought my 6 yr old the Fuji FZ10. The reason I chose it was two fold - first the zoom is built in, and in my experience the first thing kids do is knock the zoom rendering it useless, so this was a feature I wanted! Second, ease of use - to turn it on you simply slide the lens cover back and then you are ready to shoot - to turn it off you slide it back so no chance of the lens being left exposed to scratches!
This camera comes in 5 groovy colours so its a great choice for kids. Im in the UK so prices are probably not comparable with US ones but I paid about £60 British Pounds which is pretty good for a decent but kid friendly camera!