The most common mistake about Miami's name is to link it to the Miami people, a Native nation of the Great Lakes region. Wikipedia's Mayaimi article says there is no connection. The two names are a coincidence of sound.
Florida's Miami comes from Mayaimi, the name of a large lake in south Florida and of the people who lived around it. Britannica says the name "probably" means "big water" or "sweet water." It may have referred to Lake Okeechobee, or to a local people who took their name from the lake. Etymonline goes further and says the origin of the Florida name is unknown, listing the early Spanish spellings Maymi (1566) and Mayaimi (1575).
The earliest source is old. Wikipedia's Miami River article says Hernando de Escalante Fontaneda, held captive in south Florida for 17 years, called what is now Lake Okeechobee the Lake of Mayaimi. He explained that it was named for its size. That supports "big." The same article says the "sweet water" meaning is widely believed but does not confirm it.
The "big water" reading comes with a caution. Wikipedia's Mayaimi article says only ten Calusa words were recorded, so the meaning is less secure for Calusa than for the other languages. This is why the origin is best called disputed.
The name then moved downstream. Wikipedia says the river has also been called the Garband, Rio Ratones, Fresh Water, Sweetwater and Lemon River. It has been called the Miami River since the Second Seminole War of 1835 to 1842.
Britannica says Spanish explorers found a Tequesta village on the site, perhaps 2,000 years old, and that the city was incorporated in 1896. WLRN reports that voters chose city status on 28 July 1896, and Wikipedia says the city was named for the river. Wikipedia alone adds that most residents first wanted to call it Flagler, after the railroad developer Henry Flagler, who insisted it not bear his name. We found no second source for that detail.
So the city's name is a Native word for a lake, carried down a river to a city. Its exact meaning remains uncertain.