We bought our dd a Nikon Coolpix L4 for her birthday. It's in the $130 range and has 3x zoom and 4 mp, I believe. The only drawback I see with it is that it does not have a viewfinder, only an lcd, whch is probably what she'd use all the time anyway, so it's not a huge deal. It has a large variety of scene modes--individual and group portrait modes, beach and snow scenes, a bunch of different landscape modes, fireworks, etc. She's having a ball playing with all those.
For a kid, I think the key features are an lcd and optical zoom. We also bought an extended warranty that will provide replacement if she breaks it!