A quick bit of self-plugging here:
I've reviewed the video aspect of the Nano in a couple of places; a general GadgetGuy review
here and a more video specific review for Australasian Camcorder which I think is still on the front page (scroll down)
here.
End self-plugging here.
The quality is fine for web and OK for most kids as they'll be having too much fun with the whole idea of filming, BUT the designed layout of the Nano is pretty awful. They will (or at least my kids did) shoot a lot of video of their thumbs when they forget that they must hold it by the edges at all times to shoot properly.
If you just wanted handheld and kid-friendly video shooting, I'd plump for something like the Flip Mino HD or Kodak Zx1 which are better designed around that. But they're lousy music players