I made a Lunar Lander for a TI-57 calculator back in the day. I remember that I entered the code and played it multiple times so it must have been kinda fun in a way. It had a loop displaying the height and speed, then pausing for a second or so. You could halt the program during the pause, enter a new thrust value, and continue. For some reason none of the other kids wanted to play. Doing the same in 8 bit binary display will be good for practicing your grasp of binary numbers :-D
Another idea would be "guess a number between 0 and 255," which you could do both ways, i.e. user needs to guess the number the computer is thinking of, or the computer guesses the user’s number.
One can have a lot of fun with the computer doing something random and the users’ imagination does the rest. E.g. a minimal horse race game where the top and bottom nibbles say how far your two horses have run. They advance randomly. You get to give them cute names, cheer them on and place bets. It’s all up to the crowd.
Or the bits all briefly flash in some random way for a minute or so, and the spectators have to guess which one flashed most often. Flip a switch and the computer will tell you.
Maybe a puzzle game where each switch toggles some of the 8 bits. In other words each LED/bit is light based on the XOR of a subset of the switches. The user tries to find the switch setting that lights up all LEDs.
Martin