There are five students named Eaton in the Mordecai school rolls I compiled in the early 1990s:
Eliza Eaton was at the school in 1813, both sessions.
Julia Eaton was at the school from early 1814 to the end of 1815.
Rebecca Eaton was at the school in 1815, both sessions.
Temperance B. (Tempy) Eaton was at the school from early 1812 the end of 1813.
And Thomas Eaton was there for one session, early 1813.
The presence of a male student named Eaton is a clue that this is a local family. Or families.... there were a lot of Eatons in Warren County!
William A. Eaton shows up in the ledger paying for Temperance Eaton in 1812 and 1813, so that's a good set of names to start with:
Temperance B. Eaton (b. 1803) has the most distinctive name of the bunch, and she turns out to be relatively easy to find online: She was the daughter of William Allen Eaton (d. 1818) and Mary Williams, and turned 10 the year she was at Warrenton. Temperance B. Eaton married a lawyer named Lunsford Long Alsobrook in Alabama in 1826, and had one son, Jacob Eaton Alsobrook, born the following year. She probably died by 1834, when her husband married his second wife, Dorothea Stone. The Mordecais mentioned her marriage to Alsobrook, in a letter from Caroline to Ellen (29 October 1826, in the Jacob Mordecai Papers at Duke): "Mr. Alsobrook came the father of the one that married Tempy Eaton, he came for Peggy & you never saw anyone more reluctant to go than she was"--so apparently a younger in-law of Tempy's was sent to Caroline's school, too.
The other Eatons are probably not all sisters to Temperance. Julia Eaton's bills were paid by a Thomas Jenkins at the end of her time there, in December 1815. The next month, John Rust Eaton was paying the bill for Miss Dortch (Betty Dortch, we met her already). Temperance B. had an older sister Rebecca C. Eaton (1797-1840)--but
she would have been 18 during her year at the school, and would it make
sense to send the older sister to school after the younger one, without
any overlap? So I don't think this is the right Rebecca Eaton. But
it's not impossible. (But just in case she's our student, she married in 1820 to
John Howson Fenner (1798-1871), and had two children, and died at
forty-three, in Halifax NC.)
And Tom Eaton was nobody's sister, of course. Definitely local, he turns up in Caroline Plunkett's reports from Warrenton after the rest of the family has moved away, in the last 1820s; "Did I tell you Tom Eaton has left his father's again he has been in town several weeks I heard he was exceedingly disrespectful to Mrs. Eaton," she tells Ellen in 1828. In another 1828 letter (Ellen to Caroline), there's mention of Tom Eaton's poor health, but that might be another Thomas Eaton?
There were a lot of Eatons in Warren County. But I'm really not having much luck finding the one specific ones who attended the Mordecai school, except Temperance. Hmmmm.