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 would be "let go through weariness, neglect."
"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.
early 15c., "servants of a household," from Latin familia "family servants, domestics collectively, the servants in a household," thus also "members of a household, the estate, property; the household, including relatives and servants," abstract noun formed from famulus "servant, slave," which is of unknown origin.
The Latin word rarely appears in the sense "parents with their children," for which domus (see domestic (adj.)) was used. Derivatives of famulus include famula "serving woman, maid," famulanter "in the manner of a servant," famulitas "servitude," familiaris "of one's household, private," familiaricus "of household slaves," familiaritas "close friendship."
In English, sense of "collective body of persons who form one household under one head and one domestic government, including parents, children, and servants, and as sometimes used even lodgers or boarders" [Century Dictionary] is from 1540s. From 1660s as "parents with their children, whether they dwell together or not," also in a more general sense, "persons closely related by blood, including aunts, uncles, cousins;" earlier "those who descend from a common progenitor, a house, a lineage" (1580s). Hence, "any group of things classed as kindred based on common distinguishing characteristics" (1620s); as a scientific classification, between genus and order, from 1753.
Latin familia often was glossed in Old English by hired, hyred "household, family, retinue" (for which see hide (n.2), and also by hiwscipe, hiwræden, hiwan "members of a family, household, or religious house," which is cognate with Old Norse hjon "one of the household; married couple, man and wife; domestic servant," and with Old High German hiwo "husband," hiwa "wife," also with Lithuanian šeimyna "family," Gothic haims "village," Old English ham "village, home" (see home (n.)). A 15c. glossary has, for Latin familia, Middle English a menge, from Anglo-French maisnie "the household, the whole attendance upon the personal establishment of the feudal lord."
As an adjective from c. 1600; with the meaning "suitable for a family," by 1807. Family values is recorded by 1966. Phrase in a family way "pregnant" is from 1796. Family circle is 1809; family man "man devoted to wife and children, man inclined to lead a domestic life" is 1856 (earlier it meant "thief," 1788, from family in a slang sense of "the fraternity of thieves"). Family tree "graph of ancestral relations" attested from 1752.
He was dressed in his best Coat, which had served him in the same Capacity before my Birth, and possibly, might be but little short in Antiquity, to the Root of his third Family Tree; and indeed, he made a venerable Figure in it. ["A Genuine Account of the Life and Transactions of Howell ap David Price, Gentleman of Wales," London, 1752]