This story has not been well built out as yet but it also ties in with the Norse movement into the Great Lakes Country from Greenland and the general collapse of the mound building culture of the Mississippi Basin.
That the Spanish did the naming is no surprise here. This clarifies the linkages.
The Surprising Origin of the words, Maya and Miami
This surprising origin for the word, Maya, was typical of the Spanish. No Florida tribe ever called itself Timucua. Timucua is the Spanish derivation of a tribe on the Altamaha River in Georgia, named the Tamakoa (Trade People in hybrid Itza Maya-Arawak.) The Tamakoa spoke a language similar to those dialects spoken by the provinces in northeastern Florida that now are called Timucua, but they detested the Spanish and moved northward away from them. In the 1700s, the Tamakoa were located in northeast Georgia near one of the sources of the Altamaha River. By then, they were members of the Creek Confederacy.
You can blame Christopher Columbus’s brother
In 1502, Cristóbal Colón (Christopher Columbus) invited his brother, Bartolomé, and son, Fernando, to command a ship in his last voyage to the New World. While sailing through the Bay Islands off the coast of Honduras, Bartolomé encountered a large Native American sea craft with at least 25 rows of oars. In his log, Bartolomé wrote that the merchant craft was from Maiam. That is the Late Medieval Castilian spelling of the word. Renaissance Period Spanish speakers spelled the word Mayam, because by then, a Spanish “y” had the “ē” sound in English. The word was pronounced, Mă : ē : ăm, not Mă : yăm.
Mayam was a province on the northwestern tip of the Yucatan Peninsula. It is now theorized by Mexican anthropologists that Mayam was originally a Toltec (Nahuatl) word, which perhaps referred to the cochineal beetle, used for dying cloth a vivid red color.
The Spanish eventually called the people in this region, Maya, and the peninsula, Yucatan. English speakers assumed that a Spanish Y was pronounced like an English Y, and thus, Mă : yă was born.
The first Spaniards to make contact with the indigenous people living around Lake Okeechobee, Florida called them the Maiami, and the lake, Lago Maiami. This name is clearly related to the original name of the northern Yucatan Peninsula. Over time the word was contracted to Maiami and then evolved to Mayaimi. The early Anglo-American settlers in southern Florida contracted the word further to Miami. The Rio Maiami became the Miami River, and ultimately the City of Miami.
During the time period when they encountered the Spaniards, the Mayaimi lived simple lives that revolved around fishing, hunting and the gathering of wild fruits and nuts. However, in the period between around 300 AD and 1150 AD, they had one of the most advanced societies north of Mexico. Their many towns contain temple mounds, ornate earthworks, ceremonial pools and large populations. The towns were inter-connected with hundreds of miles of canals and raised causeways. This is a trait that they shared with the people living in the Yucatan Peninsula. The Mayaimi also used the artistic symbols, associated with the “Mississippian Mound Builders” hundreds of years before they appeared in other parts of the Southeast. Around 1150 AD, some catastrophe, perhaps a Class Five hurricane or a plague, wiped out most of the Mayaimi population and caused the abandonment of the large towns. Their tribal name permanently disappeared from the maps after Great Britain took control of Florida in 1763.