"short hair," 1680s; attested 1570s in sense of "a horse's tail cut short," from earlier bobbe "cluster" (as of leaves), mid-14c., a northern word, perhaps of Celtic origin (compare Irish baban "tassel, cluster," Gaelic babag).
The group of bob words in English is of obscure and mostly colloquial origin; some originally were perhaps vaguely imitative, but they have become more or less entangled and merged in form and sense. As a noun, it has been used over the years in various senses connected by the notion of "round, hanging mass," and of weights at the end of a fishing line (1610s), pendulum (1752) or plumb-line (1832). The hair sense was revived with a shift in women's styles starting in 1918 (when it was regarded as a sign of radicalism) and the modern noun meaning "a bobbed hair style" dates from 1920.
In the latter years of the decade [1920s] bobbed hair became almost universal among girls in their twenties, very common among women in their thirties and forties, and by no means rare among women of sixty .... Women universally adopted the small cloche hat which fitted tightly on the bobbed head, and the manufacturer of milliner's materials joined the hair-net manufacturer, the hair-pin manufacturer, and the cotton goods and woolen goods and corset manufacturers, among the ranks of depressed industries. [Frederick Lewis Allen, "Only Yesterday"]
Related words include bobby pin, bobby sox, bobsled, bobcat.