For an EeePC, I'm shocked. My HP dv9700t gets hot enough that it feels burning (not actually hot enough to burn, but really close, the metal connectors on the side are too hot to touch after gaming), but it's a desktop replacement laptop with a dedicated GPU. I would use PC Wizard or something to check CPU temperatures and GPU temperatures. If you're going to play games, use a cooling pad.
Then again, young children have more sensitive skin, so while my laptop doesn't burn me, it may easily burn a young child. I would recommend a cooling pad of some sort. Also, never use your laptop on a soft surface (on the carpet, on a couch or bed, etc) because these block the air intake vents often located on the bottom of the PC, greatly reducing the air that the fan pushes through the heatsink. Not only is it a potential health and injury risk, but running your PC at those kinds of temperatures could damage your GPU or CPU with extended use.