People are dying because their kidneys have failed, and you probably have an extra one: should you donate it? A dialysis machine can do most of the work of a kidney, but it's not as good. A donated kidney gives someone an average of 10 additional years.  (Some of them reject it right away, others live for years before having problems.) This is pretty good: if you volunteer to be a kidney donor and are selected as a match, you give someone an average of another 10 years of life. 
A kidney donation is a surgery, however, and does have some risk for you. It's hard to calculate the risk because people who are selected for kidney donation are a weird group: they're heavily screened (so they tend to be healthier) but they're likely to be related to the person they're donating to (so they tend to be at risk for kidney problems themself). An ideal study on the dangers of kidney donation would compare donors with people who were going to donate and then didn't for some nearly-random reason. I can't find such a study, but I've heard people give odds of around 1:3000 for dying due to donating. You also spend several days in the hospital, miss some weeks of work, and take several months to fully recover.  This sounds like a lot, but giving someone 10 years of life is a lot too.
The question isn't so much "does this do good," however, but "does this maximize the good I can do for the amount of sacrifice I'm willing to accept?" GiveWell estimates that each $1600 given to the AMF for antimalarial net distribution prevents the death of someone with an average life expectancy of maybe 40 years . So another way to give a stranger 10 more years of life is to give $400 to the AMF. Kidney donation is in the same ballpark as this, in terms of the ratio of benefit (to others) to cost (to you) , but even ignoring the risk of death  I would pay that $400 just to avoid the hospital and recovery time.
Update 2012-06-22: BDan and Alexander Berger have convinced me I was wrong to write off chains. At least right now there are more possible chains than kidneys to start them, and they can be quite long. My updated estimate is that this gets you a 3x multiplier on the value of a donated kidney, bringing the $400 charity equivalent to $1200. Which is still less than the $10-$100K at which I think I would sell a kidney, but substantially higher.
(I also posted this on my blog)
 Comparison of mortality in all patients on dialysis, patients on dialysis awaiting transplantation, and recipients of a first cadaveric transplant. (1999), table 3. This is from a longitudinal study on 23,275 people who got kidney transplants from cadavers and had never had one before. In our case we're interested in live donation, but I would guess it's about the same.
 It's possible that you could do better than this if your donation to a stranger set off a chain of donations. The idea is that there are lots of people who would give their kidney to a friend of family member if they could, but are for whatever reason incompatible. You can make donor-donee-donor-donee chains where it just takes one altruistic person to set the whole thing off. (Cycles are also possible, but they don't need someone to get them started.) The question is, though, how likely are these chains to happen anyway? And how long a chain can you reasonably expect? My understanding is that these are quite rare, with most altruistic kidney donors not starting one. I'm going to ignore chains.
 Someone who gave their kidney wrote in a Guardian article: "I was in hospital for four days--the worst part of the whole ordeal was removing the catheter. In just two weeks I was back at work (the full recovery is supposed to be three months)."
 I would love to find a GiveWell estimate of what life expectancy for the people who don't die of malaria because of a net, but I can't find one. So this "40 years" is a guess after looking at life expectancies for different ages in the countries where the AMF works.
 There are huge costs to both dialysis and kidney transplant, so as a health intervention distributing antimalarial nets [makes much more sense]. From the perspective of a prospective donor, however, I don't think this matters.
 You might think that if you're someone who gives a substantial fraction of their earnings to effective charity, a 1:3000 risk of death would be pretty bad because you'd be able to help other people much less. But 3000 is really big: if you're giving $20K/year and have an expected additional 30 years of working life, a 1:3000 chance of not being able to do that has an undiscounted expected value of only $200 less donated. 
 $20K * 30 years * 1:3000 = $200