The Wet Bandits were a duo of criminals, comprised of Marv Murchins and Harry Lyme. Marv was solely responsible for the name, after leaving a calling card of leaving the water running at every house they robbed.
After escaping prison, Marv decided that they become the Sticky Bandits after wrapping his hand in tape to steal money from a donation bucket.
Behind the scenes
While a version of Marv appears in Home Alone 4 (which shares a loose connection to the original two films), the terms "Wet Bandits" and "Sticky Bandits" are never used as Marv ditches Harry and partners up with his sidekick/wife Vera and mother Molly.