This is an Americanism -- and a regional one to boot, though the past few generations it's sometimes been heard outside of the Gulf Coast. Adopted in English in the 1840s, from Louisiana French, for it was the custom in New Orleans that the most favored customers of a merchant would get a small extra item or bonus, like a couple extra chiles or a piece of candy. Despite the appearance, it's not a French root, nor Italian for that matter, for it was adopted from American Spanish la ñapa, the addition (and according to Mark Twain, in New Orleans, the custom originated in the Spanish Quarter), in turn adopted from Quechua yapa, that which is added -- there being a similar custom in the Andes. By way of example, Twain's account from Life on the Mississippi:
When a child or a servant buys something in a shop — or even the mayor or the governor, for aught I know — he finishes the operation by saying “Give me something for lagniappe”. The shopman always responds; gives the child a bit of licorice-root, gives the servant a cheap cigar or a spool of thread, gives the governor — I don’t know what he gives the governor; support, likely.---L.