There are three girls named Eastham in my student rolls for the Mordecai school. Anne, Eliza, and Mildred Eastham are almost certainly sisters, all from Halifax County, Virginia, all with James Eastham as the adult on their account. I have Anne and Mildred (Milly) arrived in mid-1814; Anne left after just one session; Mildred stayed for most of the next two years, with their sister Eliza joining her. Mildred and Eliza both finished at the school at the end of 1816. James appears in the school ledger through during sessions.
Either Milly or Eliza was ill during August 1816, requiring a visit from their talkative father: "Miss Eastham I hope has recovered before her father's anecdotes are exhausted. And I congratulate you on having a visitor that could talk." (Solomon to Ellen, 23 August 1816, Southern Historical Collection) Julia wrote about the same visit to Samuel Mordecai: "The best news I can give you is that Miss Eastham, her talkative & goodhumoured father & mother left us on Friday. She was much better & will I hope soon recover. Her father must I think be a good man, he has at any rate a very tender heart, he bid us farewell with tears in his eyes & was so much affected that he could hardly speak." (Julia to Samuel, August 1816 [und.], Southern Historical Collection)
A James Eastham was deputy sheriff of Halifax County in 1815; there are a lot of Easthams in Halifax County, but he seems like a good candidate for starters. The same man was also the county surveyor in 1810. But his name mostly turns up in legal documents, no family history I can see.
I see a Mildred Hardeman Eastham (1805-1857), who was born in Virginia, married Alfred Hicks Rose (a fellow Virginian) in 1828, had seven children, and died in 1857 in Tennessee (here's her grave). Her dates are perfect, and we know that a lot of Mordecai-connected families moved west to Tennessee in the 1820s.
Now, here's a thought: What if Ann and Eliza are the same person? Their times at the school don't overlap, and if anything it makes more sense if she's one person--it means two sisters, Ann Eliza and Mildred, who were at the school simultaneously, arriving in mid-1814 and finishing in 1816. I found an Ann Eliza Eastham (1803-1881) who was born in Halifax Co. Virginia, married Thomas J. Spencer in about 1819, had two children, was widowed very young, and died in 1881. Her dates are perfect for a Mordecai student.
I have no evidence at hand that Mildred and Ann Eliza were sisters, or were Mordecai students--only their dates and place of birth, really. But I'm intrigued at merging Ann and Eliza Eastham into one student. Makes more and more sense as I think of it.... any clues from Virginia family historians out there?