Weeks is a common British surname, originating as far back as 1084. It is a patronymic of the Middle English word Wikke, which means 'battle, war'; a topographical name deriving from Wick, 'small, outlying village of a larger township', from residence on or near a dairy farm, or an occupational name for a worker at such a place; or an Americanization of the Scandinavian name Vik, 'small bay, inlet'.
The addition of a final "s" to topographical and locational surnames was a usual medieval practice, denoting one who was resident at a place, rather than from it.
Also, the name has variant spellings, based upon the fact that in medieval times, spelling was based upon how a word sounded. So people with these names are technically Weeks as well: Weekes, Wicks, Weech, Week, Weeke, Wich, Wych, Weetch, Wick, Wickes, Wix, Wike, Witch, Wykes, Whick, and Vik.