How detailed do they want the "mechanical" part of this project?
You could build an entirely manual version of Taito's Ice Cold Beer fairly easily, with the tools you have access to.
Instead of using the joysticks to raise/lower the sides of the rod, you would just use handles.
I don't know if that would qualify as a mechanical toy though, since there aren't any real mechanical features.
I've thought about building a golf-themed version of that where you have to get the ball to hole 1, then 2, etc... with a ball counter across the top, to count the number of balls you drain to make it through the nine holes.
If I were doing the project I'd probably head down the Chexx hockey path.
If you did do a pinball-based project, Flipper Football would probably be a good pin to base it on.
You could put multiple buttons up the sides of the machine to run various flippers to advance the ball to an enemy goal protected by gravity based knockdowns.
Another off-shoot of that would be a battle flipper type game.
You could have you and an opponent trying to shoot various targets at the peak, or just have a five ball battle to see who drains the least balls on their side, with appropriate obstacles to work past.
And lastly, at least for the moment, would be a pound and shoot basketball game.
The kind where there are holes in the playing field, and the ball stops on them until you pound a lever to send it skyward toward the opposing basket.
Ping-pong balls would work good for that one.