Asus Eee Desktop comes to mind.
I believe those have ION so at least they're half-decent in
games.
Then again..
If you want something with more beef..
Athlon II X2 240 + M4A78L-M LE + 4GB DDR2 and a 500GB HDD in a
CM 341 chassis and one of those 300W OEM PSUs from the likes of
Delta, HIPRO and Lite-ON.
All these parts can be found used online for a very low
price.
And don't worry, people aren't likely to abuse hardware...
Except for one seller that I saw on VR-Zone.
Audaciously putting up a dead Biostar motherboard on sale for
$60.
He himself states it's dead, and 'available for those who want
to try and fix it for $60'.
It's not even a very new board, and neither is it high end.
Anyway, OP; do consider getting a nettop with a decent-sized
screen like a 19" widescreen panel.
LG1943SE's are pretty affordable, and they'll suffice.
I'm afraid a small screen on a netbook is going to put a strain
on your kid's eyes.