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Old English læppa (plural læppan) "skirt or flap of a garment," from Proto-Germanic *lapp- (cf. Old Frisian lappa, Old Saxon lappo, Middle Dutch lappe, Dutch lap, Old High German lappa, German Lappen "rag, shred," Old Norse leppr "patch, rag"), from PIE root *leb- "be loose, hang down."
Sense of "lower part of a shirt" led to that of "upper legs of seated person" (c.1300). Used figuratively ("bosom, breast") from late 14c.; e.g. lap of luxury, first recorded 1802. From 15c.-In 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]That this is pleasure and not torment for the client is something survivors of the late 20c. will have to explain to their youngers.
"take up liquid with the tongue," from Old English lapian "to lap up, drink," from Proto-Germanic *lapajanan (cf. Old High German laffen "to lick," Old Saxon lepil, Dutch lepel, German Löffel "spoon"), from PIE imitative base *lab- (cf. Greek laptein "to sip, lick," Latin lambere "to lick"), indicative of licking, lapping, smacking lips. Meaning "splash gently" first recorded 1823, based on similarity of sound. Related: Lapped; lapping.
"to lay one part over another," early 14c., "to surround (something with something else)," from lap (n.). Figurative use, "to envelop (in love, sin, desire, etc.)" is from mid-14c. The sense of "to get a lap ahead (of someone) on a track" is from 1847, on notion of "overlapping." The noun in this sense is 1670s, originally "something coiled or wrapped up;" meaning "a turn around a track" (1861) also is from this sense. Related: Lapped; lapping; laps.