RaspberryPi *is* a cheap linux machine. And it is designed for children to program. It uses several different Linux distros compiled for the ARM processor from Broadcom. I have one. Kids love them. Check out Raspberrypi.org.
Your children need to learn how to program a computer and what programming is. If not we will all become a nation of users. A computer is more than an iPad that you can't even change the battery in.