Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
WITH
As described in historical dictionaries
Patronymica Britannica (1860)
WITH. A local termination occurring in such names as Sopwith, Skipwith, Beckwith. Worsaae derives it from the Danish, and says it means a forest. It is more likely identical with north, which see.