There was a student named Eliza Burton on the rolls at the Mordecai school, attending for both sessions in the year 1810. There's no adult's name or hometown attached to the listing in my dissertation appendix. And I can't find any mention of her in the Mordecai correspondence.
To add to the complications of finding this student, "Eliza Burton" isn't a very unusual name; there's an Eliza Burton Conley in Wikipedia for example (who sounds very interesting, but she's not the Mordecai student). But let's look anyway...
There are a lot of Burtons in the Virginia/NC border region around 1800. The most likely candidate for a family connection is Col. Robert Burton (1747-1825) of Granville County, NC. He was politically connected and had business associations with several other men who had Mordecai school connections; his wife Agatha Williams was kin to the Bullocks (already mentioned as having Mordecai school connections). Robert and Agatha had a daughter Eliza Williams Burton (1795-1859), their eleventh child. She married a Scotsman, James Anderson (1796-1874), and was buried in Henderson, Vance County, NC. (A photograph of her gravestone is here, and a photograph of James Anderson's gravestone in the same cemetery.) This Eliza Burton's son Robert Burton Anderson (c. 1833-1889) became a Presbyterian minister and a teacher at a South Carolina "female college," and his papers are in the Southern Historical Collection at UNC-Chapel Hill.
There's a good chance she's the Mordecai student--the age and location and social status are all perfect for a girl to be at the school in 1810. But I'd still love to find a stronger tidbit of information to secure that match. Maybe there's a letter in the Robert Burton Anderson papers, recounting his mother's story.