Etymology
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cockroach (n.)

popular name of a troublesome, voracious insect genus, 1620s, folk etymology (as if from cock (n.1) + roach; compare cockchafer) of Spanish cucaracha "chafer, beetle," from cuca "kind of caterpillar." Folk etymology also holds that the first element is from caca "excrement," perhaps because of the insect's offensive smell.

A certaine India Bug, called by the Spaniards a Cacarootch, the which creeping into Chests they eat and defile with their ill-sented dung [Capt. John Smith, "Virginia," 1624].
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roach (n.1)

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).

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beetle (n.1)
insect of the order Coleoptera, Old English bitela "beetle," apparently originally meaning "little biter, biting insect," from bitel "biting," from Proto-Germanic *bitan, from PIE root *bheid- "to split," with derivatives in Germanic referring to biting.

By normal evolution it would be *bittle, but it seems to have been influenced by beetle (n.2). Sometimes applied to soft insects, as black beetle, an old name for the cockroach. As a nickname for the original Volkswagen car, 1946, translating German Käfer.
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