c. 1400, also raindere, reynder, rayne-dere, genus of deer inhabiting the arctic regions of Europe, from a Scandinavian source such as Old Norse hreindyri "reindeer," with dyr "animal" (see deer) + hreinn, the usual name for the animal in Old Norse, from Proto-Germanic *khrinda- (source also of Old English hran "reindeer;" German Renn "reindeer," which was altered by folk etymology influence of rennen "to run;" and Swedish renko "female reindeer," with ko "cow" (n.)).
Watkins has this from PIE *krei-, from root *ker- (1) "horn; head," with derivatives referring to horned animals (both male and female reindeer have horns; those of the male are remarkable), and thus perhaps cognate with Greek krios "ram" (see kerato-). Older sources connect it to words in Lapp or Finnish (raingo). French renne, Spanish reno, Italian renna ultimately are from Germanic.
Larwood & Hotten ("History of Signboards") write that the 1670s London tavern sign of the ranged deer "was simply intended for the Reindeer, which animal had then just newly come under the notice of the public; their knowledge of it was still confused, and its name was spelled in various ways, such as: rain-deer, rained-deer, range-deer, and ranged-deer."
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."