What is the the origin of 12,500 pounds in the FAA's definition of large aircraft?
The answer by pithblot is more or less correct. The 12,500 lb. limit was chosen as an approximate halving of the DC-3 gross takeoff weight.
Consider the following quote:

As noted earlier, when first promulgated, the 12,500 lb. weight limit
could be understood in light of the general use of the DC-3, an
aircraft with a maximum gross takeoff weight of 25,000 lbs. and
carrying from 21 to 25 passengers in airline service... The imposition
of a 12,500 lb. gross weight maximum, by effectively limiting the
number of passengers that could be carried as well, did indeed protect
the local service carriers' interests.

(U.S. House of Representatives, 94th Congress, 2nd Session, 1976, pg. 48)
References:
U.S. House of Representatives, 94th Congress, 2nd Session. (1976). The future of aviation: Hearings before the subcommittee of aviation and transportation R. &amp; D. of the committee on science and technology [no. 82]. Washington, D.C.: U.S. Government Printing Office. Retrieved from Google Books.