c. 1400, also raindere, from a Scandinavian source such as Old Norse hreindyri "reindeer," from dyr "animal" (see deer) + hreinn, which by itself was 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;" Swedish renko "female reindeer," with ko "cow" (n.)).
This is probably 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."