A trivia game could work quite well. The remote is limited, but ideal for trivia. The four arrow buttons could be used to pick your answer. The center button could be used to "lock in" an answer, or it could be used if there are five choices.
This would mean repurposing some of the buttons in the application, however. The up arrow would normally exit from the current title, but this would be used to select one of the answers. The play, FF and rewind buttons would not be needed for this application, so the play button could be repurposed to exit from the current game.
Basically, you would want a live stream of the game. You would want the box to highight your answer, show the score and time remaining, and then show the correct answer. You could, just like bar trivia play with players around the world. Advertising could be displayed between games. If users created an account and linked their box to it, they could compete with users around the world.