They're called "Netbooks" and they're a big thing right now.
However, they're not meant to be used as a "normal" computer. They often have very limited disk space, and the screens are just too small.
They make "Nettops" so to say, really small and cheap desktops, maybe get one of those instead?
The MSI Windbox was just announced, it's a VESA-mounted box that mounts to the back of a monitor, so your computer basically becomes an iMac, but much cheaper.
Also, Asus makes a similar unit that's already out.
You'd just have to buy a monitor.
But keep in mind, Intel's Atom CPUs are pretty damn slow....and don't even bother with the Linux netbooks, they just don't work as a main PC.
Anyway, I'd scratch the entire nettop/netbook idea and just get a standard PC
For $400 you can get a pretty good PC:
For anything but gaming, that thing will blaze, and if your kid wants to play some games, just slap a 4670/4850 in there instead of the 3450.