Small hinges for reusable domino rally
I think it's going to be tough to find 1000 hinges of any usable quality for a price that you will like. Even if you find hinges for $0.25 each, that's still pretty expensive.

I would look at using a piece of fabric or webbing as the hinge. A couple dozen feet would be good for the whole project. If you glued the webbing tight to the board and tight to the domino in a sharp 90 deg angle, the width of the game piece should keep it from falling away from the webbing and will keep it in a upright position. It should be able to fall towards the webbing with no problem.

I would set up a small test of a dozen pieces though before committing to any method.

You might even want to cheat a little and bevel the bottom of the domino just a touch so that it leans away from the direction it needs to fall. This will help with resetting the game board.