badger (n.)
type of low, nocturnal, burrowing, carnivorous animal, 1520s, perhaps from bage "badge" (see badge) + reduced form of -ard "one who carries some action or possesses some quality," suffix related to Middle High German -hart "bold" (see -ard). If so, the central notion is the badge-like white blaze on the animal's forehead (as in French blaireau "badger," from Old French blarel, from bler "marked with a white spot;" also obsolete Middle English bauson "badger," from Old French bauzan, literally "black-and-white spotted"). But blaze (n.2) was the usual word for this.

Old English names for the creature were the Celtic borrowing brock; also græg (Middle English grei, grey). In American English, the nickname of inhabitants or natives of Wisconsin (1833).
badger (v.)
"to attack persistently, worry, pester," 1790, from badger (n.), based on the behavior of the dogs in the medieval sport of badger-baiting, still practiced in late 19c. England as an attraction to low public houses. Related: Badgered; badgering.
A badger is put into a barrel, and one or more dogs are put in to drag him out. When this is effected he is returned to his barrel, to be similarly assailed by a fresh set of dogs. The badger usually makes a most determined and savage resistance. [Century Dictionary]