snipe hunt
1. A practical joke in which the victim is sent on a hunt for a fictitious animal called a snipe. When I was in the Cub Scouts, the youngest members were always sent out on a snipe hunt by the counselors on the night of our first big campout.
2. By extension, any hunt, search, or quest for something that is fictitious, non-existent, elusive, or illusory. The president has pledged to make universal healthcare a reality, but I think he's really just on a snipe hunt.