I have a student named Eliza Callagham (or Callaham, or possibly Callaghan or Callahan) on my Mordecai school roster, listed as being from Halifax VA, kin to James Callagham, attending the school just one session (the second half of 1815). James is shown in the ledger paying for Eliza's tuition in July 1815; otherwise I don't think the Callaghams appear in the Mordecai materials.
Seems like the name might also be Calliham... this family tree finds several spellings used interchangeably by Southern Callahams/Callihams in the 1700s and 1800s. That tree includes a James Callaham (b. c. 1773) who married a Mary "Polly Smith" in 1797, in Halifax Co., VA. That puts them just exactly in the right place and time to be the parents or at least guardians of this Mordecai student. Beyond that listing, the trail seems to run out--I can't find Eliza herself. But maybe someone else is more creative in their spelling and searching?