A snipe hunt was a catchall term for a practical joke wherein more senior personnel would send newbies to do things such as search for nonexistent items.
In 2366, Data related how William T. Riker had taught him how to hunt snipe (a fictitious animal) while USS Enterprise was at Starbase 12 earlier that year. The procedure involved Data holding a bag and calling out to the snipe. Riker had apparently also pranked Wesley Crusher in similar manner at some point, and was forced to come clean about the joke when he burst out laughing mid-explanation. Data remarked, "At least that would explain why nobody has ever seen [a snipe]." (ST novel: Federation)