a shortened form of cockroach, on the mistaken notion that it is a compound, attested by 1830.
In contemporary writing the shortening sometimes is credited to a polite desire to avoid the sexual connotation in the first syllable of the full word, especially among Americans, but this seems to be another English fiction and early uses typically are in natural history publications.
The Translator must ask pardon of any American lady, into whose hands this book may by chance fall, for making use of so vulgar a term. "Cock-roaches" in the United States, as we are told by one of the numerous English travellers through that country, are always called "roaches" by the fair sex, for the sake of euphony. [B.D. Walsh, footnote in translation of "The Acharnians," 1848]
The meaning "butt of a marijuana cigarette" is recorded by 1938, perhaps from resemblance to the insect but rather this might be a different word entirely. Related: Roach-clip (by 1968).
common small freshwater fish of northern Europe, late 12c., from Old French roche (13c.), a name of uncertain origin, perhaps from a Germanic source (compare Middle Dutch roch, Low German ruche). Applied later to fish in North America that resemble it.