You're both wrong and right. It's not Anglo-Saxon, its
Norman, but it's from
Norman French that we get the meaning "Cottager" — Cottrell, with English variant Cotterill (among others): status name for a cottager, from Old French coterel, a diminutive of cotier ‘cottager’ (see Cotter 2).
Source: Dictionary of American Family Names ©2013, Oxford University Press