The first elevators in English were inside the human body. The word named muscles long before it named a box on a cable.
Elevator is a Latin agent noun meaning "one who raises up." Etymonline traces it to elevare, "to lift up, raise," and figuratively "to lighten." That verb joins ex-, "out," to levare, "to lighten, raise," which comes from levis, "light" in weight. At root, to elevate something is to make it light.
English first used the word in the 1640s for muscles that raise a part of the body. Etymonline dates its use for a mechanical lift to 1787, originally for lifting grain.
So the word, and the machine, were already old when Elisha Otis came along. Hoisting platforms had existed since antiquity, according to the Great Patent site. Their weakness was the rope. If it snapped, the platform and anyone on it fell.
Otis solved that problem, not the lifting itself. In 1852, working at a factory in Yonkers, New York, he designed what he called a "safety hoist." Britannica describes it as the first elevator with an automatic device to stop it falling if the rope or chain broke. He sold his first machine, a freight elevator, in September 1853.
The famous moment came in 1854 at the Crystal Palace exhibition in New York. Otis rode a platform high above the crowd and had the rope cut. The platform dropped a few inches and stopped. He reportedly told the audience, "All safe, gentlemen. All safe." Britannica confirms the demonstration but not the quotation.
The first passenger safety elevator followed on March 23, 1857, in the E. V. Haughwout store on Broadway. Otis died in April 1861, months after receiving a patent for "Improvement in Hoisting Apparatus." His sons Charles and Norton ran the business, which became the Otis Elevator Company.
The word later climbed into other phrases. Etymonline dates elevator music, bland background music, to 1963.