"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 (such as Danish skrubbe "to scrub"), probably ultimately from the Proto-Germanic root of shrub, used as a cleaning tool (compare 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.
scrub (n.1)
late 14c., "low, stunted tree," variant of shrobbe (see shrub), perhaps influenced by a Scandinavian word (such as Danish dialectal skrub "a stunted tree, brushwood"). Collective sense "brush, shrubs" is attested from 1805. As an adjective from 1710. Scrub oak recorded from 1766.
Transferred sense of "mean, insignificant fellow" is from 1580s; U.S. sports meaning "athlete not on the varsity team" is recorded from 1892, probably from this, but compare scrub "hard-working servant, drudge" (1709), perhaps from influence of scrub (v.).
scrub (n.2)
"act of scrubbing," 1620s, from scrub (v.). Meaning "thing that is used in scrubbing" is from 1680s.