Hello there, I'm sure you'll get loads more replies, now that folks are back at work and can surf without being nagged
Anyhow, I'm no expert on digital cameras, but I do have a 10yr old.
I bought myself a Minolta X20, which she has been using quite happily. Spec wise, it's a 2.1MP camera, with a 3x Optical Zoom.
It's small, has an internal zoom mechanism which means there are no bits sticking out to get broken, and gaps to get 'gunk' tipped in them. It takes SD/MMC, and runs on 2 standard AA batteries. It can take movies (which seems to delight the kids), and you can still use the zoom function while videoing. This model also films with sound, although the later models (x21, x31 and Xg) don't have sound. It's limited in it's choice of picture sizes/quality, but I picked up a couple of cheap 256mb SD cards, and just leave it on it's highest setting (1600 x 1200 I think), and we've not had a problem.
Also, there is no optical viewfinder, only the screen, which I know put's some people off, but my kids certainly prefer to use the screen anyway, and the fact it runs off AA batteries, means that power isn't an issue.
This is a discontinued model, and can be picked up for about £90 or so, but as it is discontinued, you might find it cheaper, or not be able to find it at all!
There are certainly far superior cameras out there, but for the money, convenience, and ease of use, I'm certainly happy with it, and, more importantly, my daughter uses it quite happily.
I bought a couple of HP Photosmart 735's for work, and I did take one home one weekend to have a play, and that was a nice camera, although my daughter thought that it was too big & bulky, she preffered the Minolta. But the picture quality of the 735 was quite impressive, and as good as the Minolta.
Hope this helps.