"sleep," Old English ræste, reste "rest, bed, intermission of labor, mental peace," common Germanic (cf. Old Saxon resta "resting place, burial-place," Dutch rust, Old High German rasta, German Rast "rest, peace, repose"), of uncertain origin.
Original sense seems to be a measure of distance (cf. Old High German rasta, which in addition to "rest" meant "league of miles," Old Norse rost "league, distance after which one rests," Gothic rasta "mile, stage of a journey"), perhaps a word from the nomadic period. Unless the original sense is "repose," thence extended secondarily to "distance between two resting place."
The meaning "support, thing upon which something rests" is attested from 1580s. At rest "dead" is from mid-14c., on the notion of "last rest." Rest stop is from 1973. Colloquial expression to give (something) a rest "to stop talking about it" is first recorded 1927, American English.