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."
Of water, "splash gently, flow against" first recorded 1823, based on similarity of sound. Figurative use of lap (something) up "receive it eagerly" is by 1890. Related: Lapped; lapping. The noun meaning "liquid food; weak beverage" is from 1560s.