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).
mid-14c., "material consisting of a compacted web or felting of vegetable fibers, commonly as a thin, flexible sheet for writing, printing, etc.," from Anglo-French paper, Old French papier "paper, document," and directly from Latin papyrus "paper, paper made of papyrus stalks," from Greek papyros "any plant of the paper plant genus," a loan-word of unknown origin, often said to be Egyptian (see papyrus).
Sense of "essay, dissertation on a topic" is from 1660s. Meaning "bills of exchange, paper money" is attested by 1722. As "paper for covering the walls of interiors," 1764. As "printed sheet of news" (a shortened form of newspaper), attested by 1640s. Papers, "collection of documents which establish one's identity, standing, credentials, etc.," it is attested from 1680s.
Paper-clip is by 1875; paper-cutter as a type of machine is by 1969. Paper-hanger is by 1796. Paper-wasp " type of wasp that builds a nest out of paper-like material" is by 1805. Paper chase is by 1856 in British English for the game of hare-and-hounds, from the bits of paper scattered as "scent" by the "hares;" the slang meaning "effort to earn a diploma or college degree" is by 1932.
Old English læppa (plural læppan) "skirt or flap of a garment," from Proto-Germanic *lapp- (source also of Old Frisian lappa, Old Saxon lappo, Middle Dutch lappe, Dutch lap, Old High German lappa, German Lappen "rag, shred," Old Norse leppr "patch, rag"), of uncertain origin.
Sense of "lower front part of a shirt or skirt" led to that of "upper legs of seated person" (c. 1300). Used figuratively ("bosom, breast, place where someone or something is held and cherished") from late 14c., as in lap of luxury (which is first recorded 1802). To drop or dump something in someone's lap "shift a burden" is from 1962. From 15c.-17c. the word (often in plural) was a euphemism for "female pudendum," but this is not the source of lap dance, which is first recorded 1993.
To lap dance, you undress, sit your client down, order him to stay still and fully clothed, then hover over him, making a motion that you have perfected by watching Mister Softee ice cream dispensers. [Anthony Lane, review of "Showgirls," New Yorker, Oct. 16, 1995]
Lap-clap was old slang for "an act of coition" (c. 1600), in warning expressions to youth often paired with lip-clip "a kiss." Also compare slang Lapland "the society of women."