Every computer with a network card has a MAC address. It's a string of 12 hexadecimal characters.
Run the getmac command from the shell. This should allow you to either
a) reserve a IP address for a specific network card, and then
configure the computer to use that IP address
b) set access limitations based on MAC address, not merely on IP address.
(there's a way to spoof MAC addresses, but it's terribly complicated and should occupy your kids' brains for some time yet.)