rest-cure n.

[sleep, repose, slumber] Old English ræste, reste "rest; a bed or couch; intermission of labor; mental peace, state of quiet or repose," from Proto-Germanic *rasto- (source also of Old Saxon resta "resting place, burial-place," Dutch rust, Old High German rasta, German Rast "rest, peace, repose"), a word of uncertain origin.

The original prehistoric signification of the Germanic noun was perhaps 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." If so, perhaps a word from the nomadic period. But if the original sense was "repose," the sense was extended secondarily to "distance between two resting places."

Sense of "absence or cessation of motion" is from late 15c. The meaning "that on which anything leans for support, thing upon which something rests" is attested from 1580s, with specific senses developing later. In music, "an interval of silence," also the mark or sign denoting this, 1570s.

At rest "dead" is from mid-14c., on the notion of "last rest, the big sleep, the grave." The roadside rest stop for drivers on busy highways is by 1970. The colloquial expression give (something) a rest "stop talking about it" is by 1927, American English.

[R]est and repose apply especially to the suspended activity of the body ; ease and quiet to freedom from occupation or demands for activity, especially of the body ; tranquility and peace to the freedom of the mind from harassing cares or demands. [Century Dictionary]

c. 1300, "care, heed," from Latin cura "care, concern, trouble," with many figurative extensions over time such as "study; administration; office of a parish priest; a mistress," and also "means of healing, successful remedial treatment of a disease" (late 14c.), from Old Latin coira-, a noun of unknown origin. Meaning "medical care" is late 14c.