According to The Random House Dictionary of American Slang (1997), the phrase "drop the hammer on" has been used as an idiom since at least 1978:
drop the hammer on to take decisive action against; "lower the boom on."
As nearly as I can tell, the inclusion of big in the OP's example does not indicate that "drop the big hammer" is a fundamentally different idiom from "drop the hammer" (in contrast to "drop the big one," for example, where "the big one" refers to a nuclear warhead); instead, I read the word big as a simple intensifier tacked onto the older and simpler phrase.
It is possible (and perhaps even likely) that "drop the hammer" evolved from "put the hammer down," a trucking term. Robert Chapman & Barbara Kipfer, Dictionary of American Slang, third edition (1995) has this entry for hammer down:
hammer down adv phr truckers by 1960 Going full speed; with throttle to the floor; =WIDE OPEN ...a herd of LA rednecks, all of 'em pie-eyed and hammer down—Esquire