The Olympics are named after a place, and it is not the mountain most people picture. Mount Olympus, home of the Greek gods, is in northern Greece. The games were held at Olympia, a sanctuary of Zeus in the western Peloponnese, a long way south.
The two names are still linked. English Olympic comes from Greek Olympikos, built on Olympos. Etymonline lists both the mountain and the sanctuary at Olympia as things the adjective referred to. Olympos itself is of unknown origin. The etymologist Beekes suggested it was a pre-Greek word that simply meant "mountain," which would explain why so many peaks carried it. The classicist Gregory Nagy lists Olympus mountains from Euboea and Lesbos to Lydia and Cyprus.
Tradition dates the first games at Olympia to 776 BCE. They were tiny. The only event was the stadion, a sprint of one length of the track, about 192 meters. The first winner on record was Koroibos, or Coroebus, of Elis, remembered as a cook and baker. His prize was an olive wreath.
Ancient writers did not agree on when the games really began. Coroebus was simply the earliest winner known to the ancient writer Hippias, so his victory became the starting point. Each later Olympiad was named after its stadion winner, which gave the Greek world its first reliable way of counting years.
That is what an Olympiad was. The word, from Greek Olympias, meant the four-year period between games, and Greeks used it as a unit of time. English had the word by the late 14th century. The adjective Olympic followed around 1600, and the plural noun Olympics, short for Olympic games, by the 1630s.
The ancient games ran for more than a thousand years. They ended in 393 CE, after 293 Olympics, when the emperor Theodosius banned pagan festivals. The modern revival held its first games in 1896 and brought the old vocabulary back with it. Even Olympian, once used for the twelve greater gods, came to mean a modern competitor, a sense Etymonline dates to 1976.