The biggest potential health benefit that a placenta can offer is serving as part of the organ infrastructure needed to harvest embryonic stem cells from the umbilical cord. Immediately after the placenta is delivered the umbilical cord us clamped on both sides (the end connected to baby, and the end connected to the placenta). One of the O.R. nurses then extracts the blood from the umbilical cord with equipment used to perform a blood test. The umbilucal cord is pierced with needle, and the embryonic cord blood then drains through plastic tubing into a special collection bag that contains additives to stabilize the blood and allow it to be FedEx'ed to a facility where it is centrifuged to separate the embryonic stem cells from the plasma. The culture is verified and then immediately placed into cryogenic storage.
It costs about $1,000 for the kit, which you simply bring with you to the hospital when baby is born. The cost of the kit includes the FedEx shipping cost, processing and first year of cryogenic storage. Each year thereafter it costs $100 to store each child's embyonic stem cells at the cryo facility. The stem cells are viable for 20 years, maybe longer. They can be used to treat disease of many kinds now, and far more in the future. If needed embryonic stem cells can be used to treat the child, a sibling or their parents. The immune cells have not developed and there is no risk of rejection.
My wife and I chose to store the embryonic stem cells of both our children. You only have one chance to collect it. And with the arcane and completely ignorant laws limiting stem cell research, it's nice to have a culture you know is compatible, and can thus find a facility with staff who specializes in a given procedure, should the need arise of course. Considering cord blood is thrown away if not utilized by the parents, I think it should be collected for resesrch purposes - as it would end the political squabnles over use of stem cells obtained from aborted fetuses, yet still provide the same high wualith cells at a volume significant enough to greatly benefit research, and to provide more samples that are available for treating those who need them.