Trampoline became a brand name, and the brand name was borrowed from a word for a diving board.
George Nissen grew up in Cedar Rapids, Iowa. Smithsonian Magazine says a circus in 1930 inspired him, when he watched aerialists land softly on their safety nets. With his coach Larry Griswold he built a bouncing frame of canvas and springs.
Sources disagree on the date. Smithsonian says the first workable prototype appeared in 1934. Wikipedia says Nissen and Griswold built the first modern trampoline in 1936. Either way it was a training aid for tumblers.
The name came from Nissen's travels, by his own account. Smithsonian says el trampolín is Spanish for "diving board." Wikipedia reports that Nissen said he heard the word on a demonstration tour in Mexico in the late 1930s and adapted it as a trademark.
The word itself was older. Etymonline and Merriam-Webster record English trampoline from 1798, from Spanish trampolin and Italian trampolino, "springboard." The Italian word comes from trampoli, "stilts," of Germanic origin. Nissen did not invent the word. He gave an old one a new job.
There is also a legend that a circus performer named du Trampolin invented the device. Wikipedia calls the story "almost certainly apocryphal" and notes that no documentary evidence supports it.
Smithsonian says Nissen and Griswold started a company in 1942. The US patent for his "tumbling device" followed in 1945. During the Second World War the US Navy used trampolines to train pilots, according to Wikipedia. The sport itself began as "rebound tumbling," and the trademarked name later became the generic word. The trademark then became generic, Wikipedia says.
Trampolining joined the Olympics at Sydney in 2000.
The result is a small irony. Nissen picked a word to make his product sound distinct, and it worked so well that everyone now uses it for every bouncy frame, whoever made it.