This tradition is newer than you might think, probably dating to the 1970s.

New Orleans's Mardi Gras celebrations centered on the French Quarter, the city's historic district, in the early twentieth century. The area had become unfashionable, as the wealthy old families moved out to get away from the immigrants moving in. The krewes, social groups that created/paid for the creation of floats for the parade, had held many of their Carnival balls in the French Opera House there, but it burned down in 1919, giving the old New Orleans families one less reason to stick around. Prostitution was made illegal in New Orleans in 1917 due to pressure from the US Navy, and the licit brothels of the regulated "Storyville" red light district were shut down; new ones were set up in the now-affordable French Quarter, giving it an even more unsavory reputation. While the Carnival balls were no longer being held there, the parades still were, so people continued to attend, but the speakeasies on Bourbon Street in the 1920s (regular bars in the 1930s) helped to bring in rowdier tourists.

But the French Quarter float parade, the huge spectacle that brought people to the area, was shut down in 1973. With bigger hotels being built and the "super-krewe" Bacchus making bigger floats and bringing in big celebrities in the 1960s, more people had been flooding the narrow, eighteenth-century streets, and the authorities were concerned that if any accidents were to occur, fire trucks and ambulance would not be able to get through. Instead, the parades moved to St. Charles Avenue, leaving a French Quarter festival that was just centered on being in a crowd - in a period that was seeing an increased frankness about and interest in sexuality and nudity, especially public nudity (like streaking).

It first started around the time that the parade was canceled with incidents of men flashing their penises at the intersection of Bourbon and St. Ann, and from the balconies of some of those newer hotels - at this point, women only made up about 25% of Mardi Gras participants. In 1975, there is said to have been a party of nudists on the second floor of a hotel that came out on the balcony and urged people on the street to reciprocate by showing their own genitalia or breasts - and threw out beads, connecting the exposure with necklaces. When the police went on strike for Mardi Gras in 1979 and all parades were canceled, there was even more riotous activity as participants had nothing to do but revel. By the 1980s, the tradition of exhibitionism being paid for with beaded necklaces was pretty well set in stone. Krewes had been tossing beads to the crowds since China's cheap plastic exports made it possible to do so, but that was random: having the opportunity to "earn" necklaces seems to have generated enthusiasm and given the beads more value.

The erasure of men being involved by showing their penises has come about in part from more women being present than in the early 1970s, when the Bourbon Street scene was largely gay men, but also from the commodification of women's public nudity through the Girls Gone Wild franchise, which only shows the women who take part, and also then caused more emphasis on getting women to disrobe - but now we are creeping into twenty-year-rule territory.

Sources:

Reason in the City of Difference: Pragmatism, Communicative Action and Contemporary Urbanism, by Gary Bridge (Psychology Press, 2005)

Bourbon Street: A History, by Richard Campanella (LSU Press, 2014)

"Contrasts of Carnival: Mardi Gras Between the Modern and Post-Modern", by Kevin Fox Gotham, from Illuminating Social Life: Classical and Contemporary Theory Revisited, edited by Peter Kivisto (SAGE Publications, 2013)

Beads, Bodies, and Trash: Public Sex, Global Labor, and the Disposability of Mardi Gras, by David Redmon (Routledge, 2014)