There was a student at the Mordecai school named Harriet Exum, for five sessions, non-continuous--she was there in 1810, 1811, and 1813. A "Capt. Exum" is mentioned in 1811 and 1813 in the school ledger, and is presumably the adult on her account.
Exum is a fairly distinctive name, but it's found as a first and a last name in Southern history, and goes back to the seventeenth century in Virginia. A Col. Benjamin Exum was a Dobbs County delegate to the state convention of North Carolina in 1776; he died about 1788, so Harriet probably wasn't his daughter, but maybe a granddaughter or niece?
Anyway, not having any luck finding a good Harriet candidate out there. She was probably from the Exum family based in Edgecombe County, probably kin to the Revolutionary colonel. Anyone have leads?