This is from the Webster Dictionary Online. My dad is from Mississippi (Columbus, as a matter of fact) and I have a cousin that is a retired school teacher there. He always told us that the name came from the early French explorers. It was a term they used for path. The definition below seems to agree with that. The Natchez were the local indians of the area, and the Trace was originally a commonly used pathway for trading with other tribes in the region.
Trace
Pronunciation: \ˈtrās\
Function: noun
Etymology: Middle English, from Anglo-French, from tracer to trace
Date: 14th century
1archaic : a course or path that one follows
2 a: a mark or line left by something that has passed ; also : footprint
b: a path, trail, or road made by the passage of animals, people, or vehicles