Willy has a non-combat pet ability (also possessed by Lil KT, Toxic Wasteling and Scorchling pets) in that he can kill almost any creature in the game classified as a "critter" (rats, cows, etc). Willy will automatically attack a random, single critter that is within a 10-yard radius, and the attack occurs once every 3-minutes (if no critter is within the attack radius then that attack is simply skipped at the appropriate time and the timer resets). The timer resets when Willy is first summoned (or unsummoned/re-summoned), and continues counting as long as Willy remains summoned. He also has a tendency to fall asleep and wake up with a start.
Note that Willy has a single eye; many players assume he is a pun on the phrase "One Eyed Willy", humorous slang for a penis. It is possible, although incredibly unlikely, that the name "Sleepy Willy" is a reference to the pirate One-Eyed Willy from the movie The Goonies.