I'd like to see the next version take the avatar concept further. They went half way by making it physically buzz in. Go the rest of the way, and make it only receive input from the avatar. Add a computer vision and speech recognition component, and make it read the board and listen to Trebek like the human players. Then, make it mobile, and put the avatar in the actual Jeopardy! studio.
EDIT: Clarified "human" players. Welcome to the future!
According the designers they considered it, but finally decided to concentrate on the question answering. OCR for this situation could probably be done in fast and reliable way, so it wouldn't make much of a difference. Speech recognition would be harder and less reliable, but redundant most of the time (though it presumably could have helped for the 1920s question).
Understandable, and I agree that they definitely tackled the meat of the problem. I think there could be some interesting problems in the computer vision aspect of it, though. Seeing and interpreting the board is subtly different than a straight OCR problem.
EDIT: Clarified "human" players. Welcome to the future!