"sleep," Old English ræste, reste "rest, bed, intermission of labor, mental peace," from Proto-Germanic *rastjo, *rasto. Original sense seems to be a measure of distance (cf. Old High German rasta "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. The meaning "support, thing upon which something rests" is attested from 1580s. At rest "dead" is from mid-14c. Rest stop is from 1973. Colloquial expression to give (something) a rest "to stop talking about it" is first recorded 1927, American English.
"remainder," early 15c., from Middle French reste "remnant," from rester "to remain," from Latin restare "stand back, be left," from re- "back" (see re-) + stare "to stand" (see stet). Related Middle English verb resten (mid-15c.) is in rest assured.
Old English ræstan, restan "to rest," from root of rest (n.1). Related: Rested; resting. Phrase rest you merry is from 1540s (God rest you merry, gentlemen, often is mis-punctuated).