"sleep," Old English ræste, reste "rest, bed, intermission of labor, mental peace," common Germanic (Old Saxon resta "resting place, burial-place," Dutch rust, Old High German rasta, German Rast "rest, peace, repose"), which is of uncertain origin, with no obvious cognates.
Original sense seems to be a measure of distance (compare 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.
"remainder, that which is left after a separation," early 15c., from French reste "remnant," from rester "to remain" (see rest (v.2)). Meaning "others, those not included in a proposition" is from 1530s.
"repose, cease from action," Old English ræstan, restan "take repose by lying down; lie in death or in the grave; cease from motion, work, or performance; be without motion; be undisturbed, be free from what disquiets; stand or lie as upon a support or basis," from Proto-Germanic *rastejanan (source also of Old Frisian resta, Middle Dutch rasten, Dutch rusten, Old High German raston, German rasten, Swedish rasta, Danish raste "to rest"), from the root of rest (n.1).
Transitive senses "give repose to; lay or place, as on a support or basis" are from early 13c. Meaning "cease from, have intermission" is late 14c., also "rely on for support." Related: Rested; resting. Resting place is from mid-14c.
"to be left, remain," mid-15c., from Old French rester "to remain," from Latin restare "stand back, be left," from re- "back" (see re-) + stare "to stand," from PIE root *sta- "to stand, make or be firm." Partially confused and merged with the other verb rest. Sense of "to continue to be" is in rest assured. Transitive sense of "to keep, cause to continue to remain" was common in 16c.-17c., "used with a predicate adjective following and qualifying the object" [Century Dictionary].
Hence the phrase rest you merry (1540s), earlier rest þe murie (mid-13c.), as a greeting, "rest well, be happy," from the old adverbial use of merry. The Christmas carol lyric God rest ye merry, gentlemen, often is mis-punctuated.