I'm planning to use an ATTiny85-based Digispark board to drive this device. It will have a nice, big, easy-to-push button for my son to use, and some software button debouncing to prevent "double-clicking". I'll use cyborg5's version of IRLib for driving the television remote codes.
I used arcade buttons for my special-needs projects when my boys were younger which made it easier for them to press the correct button.
I've also made quite a few IR enables projects so if you need any help just ask and I'll help with what I can. Have made remote controls in the past so the standard TV IR protocol(s) is something I already know.