Complete computer system
I need the cheapest possible "computer" where I can teach programming to 1000's of kids. A computer is 1) a computer with networking (Omega2 or Raspberry PI); 2) keyboard and pointing device; 3) a screen.
Teaching computer programming will be much cheaper, simpler and more widespread when a complete computer can be purchased. Omega2 goes a long way towards that goal.
@Ed-Bradford Given your requirements (particularly the need for screen, keyboard and pointing device) I don't think it would be viable to go with the Omega 2. Though it is more expensive, I think the Raspberry PI 3 would be more suitable.
You might want to consider a Pi-top: https://www.sparkfun.com/products/13896
Ed, even a phone could be a complete computer system, but still very expensive (for a nice screen and imbedded keyboard). If you want to go the inexpensive route, you have to compromise with the kind of devices that the computer will have. There are tons of single board computers out there, all with their own characteristics. Using the onion, you can add the OLED as display, and a few buttons as input, and off you go. It's very limited and requires imagination, but it certainly is possible.
Kit, I love your Pi-Top suggestion. This one could be an excellent gift suggestion as well.