quiet
Quiet and while come from the same PIE root — *kʷyeh₁-, "to rest, be at rest." Latin specialized the root into the state of rest: quies ("rest, repose"), quiescere ("to come to rest"), quietus (past participle, "at rest") — giving English quiet, quiescent, acquiesce, requiem (from requies, "rest again," the Catholic Mass for the Dead), coy (via Vulgar Latin *quetus and Old French coi), and the legal-discharge cluster quit / quite / requite / acquit / quittance — released from obligation as a kind of being-at-rest from it. Germanic specialized the same root into the time-during-which-rest-happens: Old English hwīl, from Proto-Germanic *hwīlō-, meant "a period of rest, a pause" — modern English while, in which the older pause-sense still survives in compounds (worthwhile = worth the pause; once in a while = once in a pause; to while away an hour = to spend it in pause) and the archaic whilom (dative plural hwīlum, "at the times-of-rest, sometimes"). Modern English received both branches and uses them for different work — quiet for the state, while for the time — but underneath they are one word saying one thing. We say a quiet while without hearing that we are saying a rest's rest.
Quiet and while come from the same Indo-European root — *kʷyeh₁-, "to rest, be at rest." The daughter languages split the labor between the two senses the root could carry. Latin took the state of rest and built a wide derivational family: quies, the noun; quiescere, the verb of entering it; quietus, the past participle. Germanic took the time of rest and built Proto-Germanic *hwīlō- → Old English hwīl, "a period of rest, a pause." Modern English received both branches and uses them for different work — quiet for the state, while for the time — but the doubling is real: when we say a quiet while we are saying a rest's rest in two languages stacked.
On the Latin side, the rest-family branched in two directions. The first kept the bodily-state sense: quietus → quiet (the adjective and noun), quiescere → quiescent (still resting, dormant), acquiescere ("to come to rest in") → acquiesce (to settle into a thing, to stop pushing against it), requies ("rest again") → requiem, the opening word of the Catholic Mass for the Dead — Requiem aeternam dona eis, Domine, "Give them eternal rest, Lord." From Vulgar Latin *quetus through Old French coi ("quiet, still") came English coy, in which "quiet posture" semantically narrowed to "shy manner."
The second direction was legal. Latin quietus acquired the sense "discharged from obligation, free of claim, released from a thing one was bound to," and Medieval Latin quitare, "to release from a debt or duty," entered Old French as quiter. From there English got the whole cluster: quit (to release oneself from a thing — a job, a habit, a fight); quite (originally "completely released from," "entirely free of," surviving in the modern intensifier quite good as "fully good, free of qualification"); requite (re- + quit, "to release back, to repay"); acquit ("to release from a charge," now the legal verdict); quittance (the receipt or deed of release). The same root that names rest also names being-released-from. The legal vocabulary is the rest-state mapped onto the human obligation: to be quit of a debt is to be at rest from it.
On the Germanic side, the root took a different direction. Old English hwīl, from Proto-Germanic *hwīlō-, meant "a period of rest, a pause, a span of time." The dative plural hwīlum ("at the times-of-rest") was the everyday adverb for "sometimes, occasionally, at certain pauses" — preserved fossil-like into modern English as the archaic whilom, "formerly," one of the only common survivals of Old English noun-case morphology in modern adverbs. Modern English flattened the time-sense, but traces of the older pause-shape are still inside the words. To while away an hour is to spend it restfully, in pause. Worthwhile is not quite "worth the time"; it is worth the pause, worth taking the breath for. Once in a while means once in a pause. Whilst is the same word with parasitic -t from analogy with against; erstwhile (erst "earliest" + while) is "of a former pause." Each compound is a fossil of the older sense.
Several English words for stillness come from PIE roots that have nothing to do with *kʷyeh₁-. Calm is from Late Latin cauma, "the heat of midday," through Greek kaûma, "burning heat" — the semantic shift is from the heat-during-which-no-one-works to the stillness itself; the word names the cause of the rest, not the rest. Still is from PIE *stel- "to put, stand, be motionless" — neighboring meaning, different root; the Germanic word for not-moving is from the standing-verb, not the resting-verb. Rest itself is from Old English ræst on a different (contested) PIE root. Silent is from Latin silēre, of obscure pre-Latin origin. Peace is from Latin pax, from PIE *peh₂ḱ- "to fasten" — originally a treaty-shape word, "the thing fastened down between parties." Five English words for the same domain, on five different PIE roots, each naming the quiet from a different conceptual angle: the cessation (quiet/while), the standing-still (still), the giving-up (rest), the not-speaking (silent), the agreement (peace).
- requiem — Latin re- + quies, "rest again"; the Catholic Mass for the Dead opens Requiem aeternam dona eis, Domine ("Give them eternal rest, Lord"); the modern sense "music for the dead" generalized from the liturgy
- acquiesce — Latin acquiescere, "to come to rest in" (ad- + quiescere); to acquiesce is to settle into a thing, to stop pushing against it
- quiescent — Latin quiescere + -ent; still at rest, dormant; technical in biology (a quiescent cell) and geology (a quiescent volcano)
- coy — Old French coi, "quiet, still," from Vulgar Latin *quetus, from Latin quietus; the semantic shift "quiet posture" → "shy manner" is documented in 14th-c. French and English
- quit — Old French quiter, from Medieval Latin quitare, "to release from obligation"; the legal sense of being discharged from a debt; modern quit keeps the older sense of releasing oneself
- quite — originally "completely released from, entirely free of"; the adverbial use developed from past-participle quit; the modern intensifier quite good preserves "fully good, free of qualification"
- requite — re- + quit, "to release back, to repay"; "I will requite your kindness" = "I will discharge a corresponding obligation back to you"
- acquit — Old French aquiter, ad- + quiter, "to release from a charge"; modern legal verdict
- quittance — Latin quietare → Old French quitance; a receipt or deed of release from an obligation; archaic, surviving in legal phrasing
- tranquil — Latin tranquillus, "calm"; traditionally placed on this root with trans- ("across, beyond") intensifying quietus; some modern reconstructions (de Vaan) find the morphology irregular and leave the PIE source uncertain — included with that hedge
- while (noun) — Old English hwīl, from Proto-Germanic *hwīlō-, from PIE *kʷyeh₁-; originally "a period of rest, a pause," flattened in modern English to "a stretch of time"
- while (conjunction) — the same word used adverbially: "while you wait" = "in the pause during which you wait"; the conjunction is the noun grammaticalized
- whilom — Old English hwīlum, dative plural of hwīl ("at whiles, sometimes"); archaic adverb in modern English meaning "formerly"; preserves OE noun-case morphology fossil-like
- whilst — Middle English whiles + parasitic -t by analogy with against, amongst; common in British English
- erstwhile — erst "first, earliest" (Old English ǣrest) + while; "of a former time"; preserves the time-sense of while as a unit-of-period
- worthwhile — modern English compound (16th–17th c.); literally "worth the period of pause"; closer to "worth taking the pause for" than to "worth the time"
- calm — different root — Late Latin cauma "the heat of midday," from Greek kaûma "burning heat" from kaíein "to burn"; the rest is named via the cause of the rest, not the rest itself
- still (adj.) — different root — PIE *stel- "to put, stand, be motionless"; the Germanic word for not-moving is from the standing-verb, not the resting-verb
- rest (n.) — different root — Old English ræst / rest, Proto-Germanic *rastō; PIE source contested (*res- "to rest" or *erH- "to be at rest")
- silent — different root — Latin silēre "to be silent," of obscure pre-Latin origin; no secure PIE etymology
- peace — different root — Latin pax, from PIE *peh₂ḱ- "to fasten"; originally a treaty-shape word, "the thing fastened down between parties"