Robert Hendrickson, The Facts on File Encyclopedia of Word and Phrase Origins (1997) gives an earlier date for "big cheese" in the sense of "important person" than either Green (cited by Brian Hooper) or Oxford Dictionaries (cited by Andrew Leach) give:
big cheese. A big cheese, for "a boss or important person," is an Americanism dating back to about 1890. But it derives from the British expression the cheese, meaning "the thing or the correct thing, the best." The British expression, in turn, is a corruption of the Persian or Urdu chiz (or cheez), "thing," that the British brought back from India in about 1840. A big cheese thus has nothing to do with cheese and should properly be "a big chiz."
Unfortunately, Hendrickson doesn't provide a citation for his "about 1890" date—or for any other date.
As for the origin of the British term the cheese, Farmer & Henley, Slang and Its Analogues (1891) offers this discussion:
Summing up the evidence, the expression—(barring a solitary reference in the London Guide of 1818, where it is referred to a bald translation of c'est une autre chose, i.e., that is another CHEESE, subsequently coming to signify that it is the real thing)—appears to have come into vogue about 1840. This contention is borne out in some measure by a correspondent to Notes and Queries (1853), I, S., viii., p. 89), who speaks of it as about "ten or twelve years old," a calculation that carries it back to the date when it appears to have started in literature. Yule, writing much later, says the expression was common among young Anglo-Indians, e.g., 'my new Arab is the real chiz,' i.e., 'the real thing,' a fact which points to a Persian origin.
According to Farmer & Henley, one of a handful of contemporaneous terms for an important person was "big bug":
BIG BUG, subs. [popular]—A person of standing or consequence, either self-estimated or in reality. A disrespectful but common mode of allusion to persons of wealth or with other claims to distinction. Variants are BIG-DOG, BIG-TOAD, BIG-WIG, and GREAT GUN.