The origin of the term is a practical joke where inexperienced campers are told about a bird or animal called the snipe as well as a usually ridiculous method of catching it, such as running around the woods carrying a bag or making strange noises.The purpose of a wide democratic snipe hunt is to keep the opposition occupied with an impossible task. It's a bonus that they get a laugh out of it.
* Is there anything that wikipedia doesn't cover?
And the best of snipe hunters can catch a snipe, too. I do seem to recollect that I was "Order of the Snipe Arrow" or something like that.
I'm pretty sure that I come from a family where we learned to convince even the people who set up the snipe hunt as a joke that there really are snipes, and that gee yes we did catch one, and yet had to let it go because the mother snipe came lumbering up and took little jimmy as hostage, and he wouldn't be here alive unless we had freed the baby snipe and drove the two of them away.