Many parents end up saving their baby’s umbilical cord, after it dries up and falls off. I recently heard of this old wives tale that if a woman cannot conceive, belief is if she consumes a piece of this umbilical cord, it would help her become pregnant. I have also heard that the umbilical cord once dry tastes rubbery. It is usually given with a banana, both to hide the taste and possibly the cord itself.
Medically, what I have heard is that the umbilical cord is a source of stem cells, similar to bone marrow. Therefore you have cases where the cord is preserved.
However, here are two bizarre stories that I found about oral consumption.