has a few hypotheses, none definitively established. I looked it up because Florida is so distant from the base range of the Shawnee. The Shawnee origin was suggested by an Indian agent as a guess, but the Shawnee had only one village in Florida, not on the Suwannee River. The Native people of the area were the Timucua and their name for the river was Guacara. One suggestion made in the late 1800s is that the name comes from the Creek tribe's word, Siwani, meaning "echo." But there is no evidence for that in the Creek language. The most accepted origin of the name is the Spanish mission of San Juan on the river.
The Shawnee belong to the huge Algonquian linguistic group of Native Americans whose lands before being displaced were primarily in Canada, New England, the upper Midwest (Minnesota, Michigan, Wisconsin), Ohio, West Virginia, Virginia, and parts of Pennsylvania. A few extended southeast around Georgia and the Carolinas. They include some well known tribal names - Massachusett, Wampanoag, Powhattan, Ottawa, Lenape (Delaware), Pequot, Mahican, Mohegan, Cheyenne, Blackfoot, Illini, Powhattan, Miami (in western Ohio), Shawnee, and many more.
The Shawnee migrated a lot, but are believed by some anthropologists to be descendents of the prehistoric Fort Ancient culture of the Ohio Valley - southern Ohio, northern Kentucky, and West Virginia. In historic times, they were in Ohio and frequently waged war raids on the Cherokee over hunting land in Kentucky. Chief Tecumseh was their leader in Ohio.
I am not familiar with Professor David Reich's works so I looked him up. According to the Harvard Gazette, he and a group of scholars, including British researchers, determined that there were back migrations to Siberia from Alaska. I read about that several years ago, when Native haplotypes were being discovered. His work has established that early arrivals to North America took a coastal route to South America, a hypothesis long held by archaeologists like James Adovasio, among others. So now we have DNA evidence of it. I disagree with his timing of 15,000 years ago, though, because Meadowcroft shows people in PA 14,000+ years ago, possibly to 20,000. Other sites in North America, south of the glaciers, also suggest older timelines.
But, as the Harvard Gazette reports, Reich says that the DNA for Native people of North America is too scarce for accurate migration tracing and dating until there is more available.
For a good overview of the early Americans, before they settled into linguistic families and tribal identities, I recommend James Adovasio's book, The First Americans.