But... there is a rather large extended family of Ballards in Virginia in the early 19th century. And Lucy was a name used frequently in that family, as was Rebecca to a lesser extent. For example, this Rebecca would be about the right age and location to be a Mordecai student in 1814:
"Larkin BALLARD and Elizabeth GAINES were married on 13 Jan 1786 in Orange County, Virginia. Children were: Hiram BALLARD, Howard O BALLARD, Henry BALLARD, Humphrey BALLARD, Rebecca BALLARD, Nancy BALLARD, Catherine BALLARD, Sarah BALLARD"We don't have any much to go on at this point, but that's what the blog is for--maybe you've got a Lucy (Lucretia? Louisa?) Ballard in your genealogy project who was born 1795-1800 in North Carolina or Virginia. Maybe she had a younger sister or cousin named Rebecca Ballard who would have been born more like 1800-1805. If so, tell us more about them!
ETA 3/28/12: I've heard from a family historian who has another very good candidate for this Rebecca Ballard--her grandmother's grandmother was Rebecca Taylor Ballard Woodcock, born 1800 (or possibly 1804) in Halifax Co. NC. That would make her 14 (or possibly 10) during her year at the Mordecai school, which is pretty typical. She married a doctor from Philadelphia in 1818, and they lived in Mobile, Alabama; they had at least four children, possibly as many as nine; she died in 1865.