"an untruth; conscious and intentional falsehood, false statement made with intent to deceive," Old English lyge, lige "lie, falsehood," from Proto-Germanic *lugiz (source also of Old Norse lygi, Danish løgn, Old Frisian leyne (fem.), Dutch leugen (fem.), Old High German lugi, German Lüge, Gothic liugn "a lie"), from the root of lie (v.1).
To give the lie to "accuse directly of lying" is attested from 1590s. Lie-detector is recorded by 1909.
In mod. use, the word is normally a violent expression of moral reprobation, which in polite conversation tends to be avoided, the synonyms falsehood and untruth being often substituted as relatively euphemistic. [OED]
c. 1300, at first found chiefly in northern England and north Midlands writing, "powerful, strong," of obscure origin, possibly from a Scandinavian source (compare Norwegian dialectal bugge "great man"). Old English used micel (see much) in many of the same senses. It came into general use c. 1400. Meaning "of great size" is late 14c.; that of "full-grown, grown up" is attested from late 14c. Sense of "important, influential, powerful" is from c. 1400. Meaning "haughty, inflated with pride" is from 1570s. Meaning "generous" is U.S. colloquial by 1913.
Big band as a musical style is from 1926. Slang big head "conceit" is first recorded 1850. Big business "large commercial firms collectively" is from 1913 (before that it meant "a profitable income in business"). Big top "main tent of a circus" is from 1895. Big game "large animals hunted for sport" is from 1864. Big house "penitentiary" is U.S. underworld slang first attested 1915 (in London, "a workhouse," 1851). In financial journalism, big ticket items so called from 1956. Big lie is from Hitler's grosse Lüge.