c. 1300, at first found chiefly in writings from northern England and north Midlands, with a sense of "powerful, strong;" a word of obscure origin. It is possibly from a Scandinavian source (compare Norwegian dialectal bugge "great man"). Old English used micel (see much) in many of the same senses.
Big came into general use c. 1400. The meaning "of great size" is from late 14c., as is that of "full-grown, grown up." The sense of "important, influential, powerful" is from c. 1400. The meaning "haughty, inflated with pride" is from 1570s. The sense of "generous" is U.S. colloquial by 1913.
Big band as a musical style is from 1926. Slang big head "conceit" is recorded by 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 is attested by 1915 (in London, "a workhouse," 1851). In financial journalism, big ticket items were so called from 1956. Big lie is from Hitler's grosse Lüge.