"stoppage, obstruction" (obsolete unless in legal contracts), late 12c., from archaic verb letten "to hinder," from Old English lettan "hinder, delay, impede," etymologically "make late," from Proto-Germanic *latjan (source also of Old Saxon lettian "to hinder," Old Norse letja "to hold back," Old High German lezzen "to stop, check," Gothic latjan "to hinder, make late"), related to *lata-, source of late (adj.), from PIE root *‌‌lē- "to let go, slacken."

Old English lætan (Northumbrian leta) "to allow; to leave behind, depart from; leave undone; bequeath," also "to rent, put to rent or hire" (class VII strong verb; past tense let, leort, past participle gelæten), from Proto-Germanic *letan (source also of Old Saxon latan, Old Frisian leta, Dutch laten, Old High German lazan, German lassen, Gothic letan "to leave, let"), from PIE *led-, extended form of root *‌‌lē- "to let go, slacken." If that derivation is correct, the etymological sense might be "let go through weariness, neglect."

The idea of slackening lies at the root of both applications of the term. When we speak of letting one go, letting him do something, we conceive him as previously restrained by a band, the loosening or slackening of which will permit the execution of the act in question. ... At other times the slackness is attributed to the agent himself, when let acquires the sense of being slack in action, delaying or omitting to do. [Hensleigh Wedgwood, "A Dictionary of English Etymology," 1859] 

He points to similar developments in French laisser "to let" from Latin laxare "to slacken," German lassen "to permit, to let," from dialectal lass "loose."

"The shortening of the root vowel ... has not been satisfactorily explained" [OED]. Of blood, from late Old English. Other Old and Middle English senses include "regard as, consider; behave toward; allow to escape; pretend;" to let (someone) know and to let fly (arrows, etc.) preserve the otherwise obsolete sense of "to cause to."

To let (someone) off "allow to go unpunished, excuse from service" is from 1814. To let on is from 1725 as "allow (something) to be known, betray one's knowledge of," 1822 as "pretend" (OED finds a similar use in the phrase never let it on him in a letter from 1637). To let out is late 12c. as "allow to depart" (transitive); intransitive use "be concluded," of schools, meetings, etc., is from 1888, considered by Century Dictionary (1895) to be "Rural, U.S." Of garments, etc., late 14c.

Let alone "abstain from interfering with" is in Old English; the phrase in the sense "not to mention, to say nothing of" is from 1812. To let (something) be "leave it alone" is from c. 1300; let it be "let it pass, leave it alone" is from early 14c. To let go is from c. 1300 as "allow to escape," 1520s as "cease to restrain," 1530s as "dismiss from one's thoughts." Let it go "let it pass, no matter" is as old as Chaucer's Wife of Bath: "But age allas Hath me biraft my beautee Lat it go, far wel, the deuel go ther with!" [c. 1395]. Let me see "show me" is from c. 1300.

"that is within, internal," 1590s, from in (adv.). Sense of "holding power" (the in party) first recorded c. 1600; that of "exclusive" (the in-crowd, an in-joke) is from 1907 (in-group); that of "stylish, fashionable" (the in thing) is from 1960.

also letdown, "a disappointment," 1768, from let (v.) + down (adv.). The verbal phrase is from mid-12c. in a literal sense "cause to be lowered," of drawbridges, etc.; figuratively by 1754.

a Middle English merger of Old English in (prep.) "in, into, upon, on, at, among; about, during;" and Old English inne (adv.) "within, inside," from Proto-Germanic *in (source also of Old Frisian, Dutch, German, Gothic in, Old Norse i), from PIE root *en "in." The simpler form took on both senses in Middle English.

Sense distinction between in and on is from later Middle English, and nuances in use of in and at still distinguish British and American English (in school/at school). Sometimes in Middle English shortened to i.

The noun sense of "influence, access (to power or authorities)," as in have an in with, is first recorded 1929 in American English. to be in for it "certain to meet with something unpleasant" is from 1690s. To be in with "on friendly terms with" is from 1670s. Ins and outs "intricacies, complications of an action or course" is from 1660s. In-and-out (n.) "copulation" is attested from 1610s.

"cessation, restraint, relaxation, intermission," 1837, from verbal phrase let up "cease, stop" (1787). In Old English the phrase meant "to put ashore" (let out meant "put to sea"). Bartlett (1848) says the noun is "an expression borrowed from pugilists."

1894, "anyone of a relationship not natural," abstracted from father-in-law, etc.

The position of the 'in-laws' (a happy phrase which is attributed ... to her Majesty, than whom no one can be better acquainted with the article) is often not very apt to promote happiness. [Blackwood's Magazine, 1894]

The earliest recorded use of the formation is in brother-in-law (13c.); the law is Canon Law, which defines degrees of relationship within which marriage is prohibited. Thus the word originally had a more narrow application; its general extension to more distant relatives of one's spouse is, according to OED "recent colloquial or journalistic phraseology." Middle English inlaue (13c.) meant "one within or restored to the protection and benefit of the law" (opposite of an outlaw), from a verb inlauen, from Old English inlagian "reverse sentence of outlawry."

1964, from in (adj.) + joke (n.).

"designed to be plugged into a socket," 1922, from the verbal phrase, from plug (v.) + in (adv.).

"quarrel, confrontation," 1905, from the verbal phrase; see run (v.) + in (adv.). From 1857 as "an act of running in," along with the verbal phrase run in "pay a short, passing visit." Earlier to run in meant "to rush in" in attacking (1815).