What type of space heater should I use in a child's room?
I'd go with a permanently installed baseboard heater with independent temperature controls (i.e. not connected to the main thermostat.)  These are attached to the walls so that it can't be tipped over. 

You can get safety covers for them which will keep little fingers out, and also make it more attractive.  

You will need to get an electrician in to do the wiring, so you might wish to install a wall mounted thermostat at the same time.