This got me thinking and perhaps an even simpler variant based on this idea would be to push just one pawn up for each side randomly, so the game would begin on move 2 with each side having a pawn on its own 3rd rank. These pawn moves need not be symmetrical, yielding a total of 8x8=64 starting positions (same as the total number of squares!). The variant could be called chess64 or something similar

This variant has the extra benefit of not cutting off all links to common theory. ECO classification is still possible, and skillful players may find clever transpositions into better known openings. Overall, it should probably appeal more to those who prefer quiet openings.
I've run a couple of short engine matches with it using a PGN opening suite I created that has all the 64 starting positions (chosen randomly, and played with reversed colors). In the first 10-game test , 5m+3s TC, Houdini 3 beat Stockfish 4 by a score of 6.5-3.5, but in the 10-game 40/10 match, SF4 won over H3 by 5.5-4.5

Regards,
CL