There's a student named Patsy Farrar on the rolls of the Mordecai school that I compiled in the early 1990s. She was only at the school for one session (the first half of 1810). I don't have a hometown, a parent name, or any other information.
First things: Patsy was generally a nickname for Martha, not for Patricia, as it often is today. So we can probably assume she was named Martha Farrar. And 1810 was early enough in the school's run that she was probably from NC or VA--not farther afield. Maybe more likely Virginia, because Farrar's Island was an early settlement in the Richmond area (patriarch William Farrar was born in Yorkshire in 1583 and died in Virginia in 1637--that's pretty early for an Englishman in Virginia).
This Martha Farrar (b. 1794) was from Rockingham County, NC. She would have been 16 while she was at the Mordecai school--older than most, but not impossible.
There may be better candidates; no reason to think the Rockingham Martha is definitely the Mordecai student. Anyone know more?