There's a student in the rolls of the Mordecai school named Eliza Fenner. She was at the school for a year, two sessions, mid-1810 to mid-1811. That's all I've got. But it turns she's not hard to find with just that bare minimum of detail.
Eliza Geddy Fenner, the daughter of Dr. Richard Fenner (1758-1828), and his wife Ann McKinnie Geddy Fenner (1769-1852). Fenner was the first president of the North Carolina Medical Society. Eliza Geddy Fenner was born in Franklin County, North Carolina, in 1799, which would make her exactly the right age, class, and location for a child at the Mordecai school in 1810.
Eliza Geddy Fenner married James Vaulx (1783-1862). They had four daughters and a son; two of the daughters died in childhood. They moved to Tennessee, where Eliza is buried (here's her Find a Grave site). Her son James Junius Vaulx became an Episcopal clergyman in Arkansas and briefly in West Palm Beach, Florida.
And Eliza Geddy Fenner Vaulx was apparently the great-great-grandmother of Arizona senator John McCain, through her granddaughter Katherine Davey Vaulx McCain (1878-1959).