I am not a Gleaves descendant but live in Mt. Juliet and have become interested in the controversy regarding the naming of the town.I have found that it was known as Mt. Juliet as early as 1839, when one of James K. Polk's political supporters wrote a letter urging him to do some campaigning here.
Julia Jennings would have been a girl of just 22 at that time - and of course the name may go back even further than the 1839 document - which makes it seem unlikely that the place would be named for her.But not impossible, of course.
An alternate theory is that it was named for an estate in Ireland of the same name, from which some area settlers may have come. We may never know the truth, but the Julia Gleaves story has been favored in local legend for a long time.