I think (definitely not my idea, Alexander discusses it in his post) that a comfortably large tax break in exchange for a kidney donation hits the sweet spot.
You can't get out of acute money problems with a tax break. Arguably still more appealing to the lower middle class than the upper class, but it seems literally impossible to reward people for the sacrifice involved in an organ donation without the marginal utility problem arising.
Still, I think it eliminates all the truly unpalatable failure modes, like impulsively donating a kidney to pay off a gambling debt. If some people choose to make an organ donation so they'll have more of their income over a five year period to invest into their children's college fund, that isn't such a bad thing.