A bath set is one of the many gifts that can be given to Miis. When given to a Mii, they will proceed to take a bath. Bathing removes any hair dye the Mii may have and calms them down if they had a fight, but a bath will not work with a huge fight.
A Mii may sometimes have a problem where they'll ask the player if they can take a bath. It can also be a traveler's problems at the campground.
Miis will sing a random song during bathing, but with lyrics as "la la la la" in Western releases, and humming in the Japanese release.