I don't think that you should be able to specify where your organs go. That's a bit fucked up. Imagine you were first on the list for an organ, find out there's one available, and then find out it's not going to you because of some stupid reason that the person gave before they passed. You're gone, it can help people. Why ruin their new shot at life?
I doubt they'd intentionally tell them that there's one available if it wouldn't go to them. I can see how computer/human error would cause a lawsuit of mass proportions if it happened though.
This does bring up an interesting debate. Wherein does the conscious lie? For horcruxes, they must put a bit of the conscious mind in each one, right (like the diary)?
I save all mine and my kids teeth in a box and when I pull them out and hold them, I feel powerful af.