What word does your mother always say?
early 14c., "name, title, or epithet added to a person's name," from sur "above" (see sur-) + name (n.); modeled on Anglo-French surnoun "surname" (early 14c.), variant of Old French surnom, from sur "over" + nom "name."
An Old English word for this was freonama, literally "free name." Meaning "family name" is first found late 14c. Hereditary surnames existed among Norman nobility in England in early 12c., among common people began to be used 13c., increasingly frequent until near universal by end of 14c. The process was later in the north of England than the south. The verb is attested from 1540s.