I have in my head a diorama set up with the bottom painted brown and the walls painted stripes like a circus tent.
I think the top of a water bottle would make an excellent sized wheel for a Ferris wheel. Poke a hole in the middle and affix the wheel to the box with a straight pin as an axle and bend it on the other side so it will stay in place.
You could make a mini racing strip also with a bottle cap as a 'car' but use 2 straight pins or needles as axles with bits of round cardboard from cereal boxes as wheels. Make a pizza pie shaped inclined wedge out of a hard old sponge. Cut out a bit of the box on the bottom, probably rectangle shaped, on 3 sides and glue the wedge to it on the outside so it would sit flat inside the box when you want it and to keep it attached. Sit the car here and when he wants to make it move pop in the wedge and the car should roll away under it.
You could make a small water dunking tank with a Tic tak box and a screw that would dump the sitter when the holding the box opening/ledge would be removed (press the 'ledge'/ opening door on top inside out so it needs to be supported by the screw. Color the water inside the box with blue food coloring.
Make a swing that can go up & down with a pulley by fixing a bit of cardboard to 2 overly long bits of fishing line or thread. Run them through an opening on the top of the box, back in again through a second hole and then wrap them around 2 bobbins sitting on push pins at the bottom of the box. To raise the swing roll the bobbins around and you've got yourself a pulley.
Ok, four is all I've got. Good luck and post a picture when you're done of whatever you guys come up with in the end. I love projects.