The tradition of throwing beads from floats began in the 1920s. That decade is associated with krewes regularly tossing strands of beads to spectators, creating the custom recognized today.
Mardi Gras throws themselves are older. During the 1870s, members of the Twelfth Night Revelers and other parade groups distributed gifts and trinkets to people along the route. The Krewe of Rex later helped develop the modern throw tradition, including beads and other specially chosen souvenirs.
The earliest beads were generally made of glass and imported from Japan or Czechoslovakia. They were more expensive and breakable than modern strands, so spectators treated them as keepsakes. Plastic beads became widespread in the 1960s because they were cheaper, lighter, and easier to throw in large quantities.
The 1920s answer refers specifically to beads becoming a regular float throw, not to the first ever object tossed at a Mardi Gras parade. That distinction explains why some histories cite the 1870s when discussing throws generally, while bead histories point to the 1920s.