The phrase “not to care a fiddlestick” means: to be wholly unconcerned; to care nothing at all.
Although our ancestors, some three hundred years ago, had high regard for the fiddle, they seemed to think of the fiddlestick, without which the fiddle could not be played, as a mere trifle, a bagatelle, something so insignificant as to be absurd.
Thus, Grose, in his Classical Dictionary of the Vulgar Tongue (1796) defined fiddlestick’s end as “Nothing.”
Washington Irving, in Salmagundi (1806-1807) is credited with the introduction of the present phrase into the literary language.