The square was laid out in 1733. It was one of the first squares set up by Oglethorpe.
the square took its permanent name in 1763, in honor of Royal Governor James Wright.
I had in mind the first images of the movie "Midnight in the Garden of Good and Evil" showing a policeman rinding a horse. The only time I saw horserider policemen, they were in Wright square.
When Oglethorpe arrived to build the town which became Savannah, he was helped by the chief of a Creek Indian tribe : Tomo-Chi-Chi who was already old (about 83 years). They signed a treaty and Tomo-Chi-Chi was considered as a friend of the first settlers. He was invited in England in 1734 and met the King and the Archbishop of Canterbury. When he died, in 1739, he was buried in 1739.
The colonists erected a grave in the square which was replaced by the current one in 1880.
The monument in the middle of the square does not seem to be the grave though there is a brass plate telling the story of the Indian chief (see below) just in front of it. I think there is a confusion and the monument was really erected in the honor of William W. Gordon a Savannah mayor. There is a rock with a plaque in a side of the square which seems to be the true grave of Tomo-Chi-Chi or at least a marker.