The name comes from an American game show. The mechanism is far older, and the mathematics was described before either.
Plinko appeared on The Price Is Right in 1983: a large vertical board of pegs down which contestants dropped flat discs, with cash values at the bottom. It became the show's best known segment and the name attached itself to the format permanently.
Japanese pachinko parlours had been running on the same principle since the 1920s — steel balls falling through a field of pins, prizes decided by where they land. Pachinko is mechanically far more elaborate; the core is identical.
Both descend from a device the Victorian statistician Francis Galton built to demonstrate the binomial distribution: a board of pegs with beads poured through it, piling up in a bell curve every time. Galton's point was that many independent left-or-right decisions produce a predictable overall shape even though each individual path is unpredictable.
That is exactly the property online Plinko monetises. The shape is guaranteed; your path through it is not.