Middle English le, leoh, from Old English hleo "shelter, cover, defense, protection," from Proto-Germanic *khlewaz (source also of Old Norse hle, Danish læ, Old Saxon hleo, Dutch lij "lee, shelter"). The original sense is uncertain; it might have been "warm" (compare German lau "tepid," Old Norse hly "shelter, warmth"), and Watkins traces it to a PIE *kle-wo-, a suffixed variant form of the root *kele- (1) "warm."
Nautical sense "that part of the hemisphere to which the wind is directed" (c. 1400) is of Scandinavian origin, from the notion of the side of the ship opposite that which receives the wind as the sheltered side. As an adjective, 1510s, from the noun. The lee shore is that toward which the wind blows. Middle English also had lewth "warmth, shelter," Old English hleowþ, with Proto-Germanic abstract noun suffix *-itho (see -th (2)). Also compare lukewarm.