"rub hard," early 15c., earlier shrubben (c.1300), perhaps from Middle Dutch or Middle Low German schrubben "to scrub," or from an unrecorded Old English cognate, or from a Scandinavian source (cf. Danish skrubbe "to scrub"), probably ultimately from the Proto-Germanic root of shrub, used as a cleaning tool (cf. the evolution of broom, brush (n.1)).
Meaning "to cancel" is attested from 1828 (popularized during World War II with reference to flights), probably from notion of "to rub out, erase" an entry on a listing. Related: Scrubbed; scrubbing.