The Random House Historical Dictionary of American Slang, A-G, has the following:
the highest level of attainment or prestige in any given area; (specif. and orig.) the most prestigious theaters or the major leagues of sports--constr. with the. Now collog. or S.E..
1910 Variety (Aug. 20) 12: The "ballyhoo: is not for vaudeville, present day vaudeville, "big" or "little time."
1910 in OEDS: The "big time," as such theatres as Percy Williams' and William Morris' are termed.
1913 Sat. Eve. Post (Mar. 29) 8: The "big time"--comprising the finest theatres in the largest cities.
New York City; (occ) Chicago.--usu. constr. with the.
1902 Cullen More Tales 78: You've got the [notion] that you'll be able to work me for the ride to the big town.
1915 Howard God's Man 189: Anybody'd think you just come to the Big Town the way you take it, Sonny.
1922 Tully Emmett Lawler 201: Just hit the Big Town, kid?
1933 Ersine Pris. Slang 18:
1. In the East, New York City.
2. In the West, Chicago.