buy them a rubbish camera, you don't want them taking better pictures than you.

My two got Kodaks off thier Grandad when they were 7 for thier birthday & Christmas. They had been scrapping over my cheap & nasty vivitar copy all summer & showing him the pics on Face-Book!
They did manage to get better pics than me a lot of the time! Little bludgers!
One of the brilliant things about digital; when I was 7, I was given a 110 instamatic that came free with a Reader's Digest offer or something.... and was chastised for moaning it needed film... then for wanting film processed... then for taking lots of out of focus or under exposed pictures of lego models, or action men, or 'wildlife' I found in the park, that was a mere dot in the frame!
Digital? Charge the batteries and let'em get on with it!
Both of mine have got quite good, to be honest; daugher has been more imaginative; early on she stole my tripod and took lots of pictures of a plastic dinasaur that she then animated

Lad is a little more technically minded.. and moans about his zoom reach and printing photo's of railways off big enough to use as scenary on his hornby layout!
Compact camera market seems to be a bit 'fast' the models changing quickly; and without going through loads of reviews hard to make any realistic reccomendations.
Tough and durable would seem sensible, and that old Vivitar copy of mine which was so masic and crudimentary as to not even have a lens would seem a good starting point... and has lasted me, seven years of fairly seriouse abuse... but..... kids break stuff. Fact of life, really... I doubt they have or ever will make a truly kid-proof camera.... but can you make a kid, camera proof?
Personally; I would look for the cheaper entry level compacts with smaller screens as they would incline to be harder to break, and lesser zoom ranges as they are more likely to be more robust. Lith-Ion battery; so I'm not nagged for an odd number of tripple-A's every five minutes... Some of the sub £50 compacts these days are pretty well specced; large displays, long zoom range; and very small. and there is a Nikon Cool-Pix in ASDA thats marked up at £49.95 that would seem to fit the bill, if I had to get them one now to do same job.... then stern warnings about looking after it; screen protector, wrist-strap, and let them loose.
More than like, its lottery whether they kill the thing, and whether any one camera is more of less likely to be killed by what they do with it.
I mean....if they decide to try sticky taping it to the top of a radio controlled car, or seeing if they can get underwater pictures of the fish in the pond.... it's probably not going to end well regardless..... so you give them the talk before hand, and they listen to you or they dont... they apply common sense, or they dont... the camera lives or it dont.
Bottom line; always told my kids that the camera is just a tool to get a picture; and better a broken camera and lots of pictures to look at, than no pictures and a pristine camera in a draw!