I gave the problem some thought while I couldn't sleep last night and came up with some model numbers: Bernardo is right that you're fighting both inertia and gravity when jumping and that inertia doesn't change. But if you look at a average poor jumper doing a jump from a standstill, he will probably crouch (moving his center of gravity -0.5m) and then jump so that his feet are 0.5m off the ground. So he will have to spend the same energy producing impulse for his jump that he has to use just to get up from the crouch (i.e. whatever it takes to move your center of gravity up half a meter).

On the moon, assuming everything else stays the same (no bulky spacesuits etc.) the work just to get up from the crouch decreases to 1/6, so he will have 1 5/6 times the work left for the actual jumping. So if the force that his muscles can generate is constant and the 0.5m that he crouched before the jump stay constant, he should end up with almost twice the lift-off velocity compared to earth. Of course that assumes that his muscels can contract fast enough to actually produce the speed.

Anyway, shouldn't the model jumper be able to jump almost 12x the heigth than on earth. Maybe my high school physics is so rusty that I messed up somewhere, but I don't think I did.

So, the better you are at jumping the less you benefit from the lower gravity, because the fraction of work you spend on just moving your ass, instead of accelerating, will be lower.