Professor Veitch in his "History And Poetry Of The Scottish Border" published in 1893 gives the date of writing to approx 1756 (possibly slightly later than Alison Rutherford's version though neither were published at that point) and names her as Jean Elliot the third daughter of Sir Gilbert Elliot, second baronet of Minto.

Jean and Jane are still interchangeable amongst older women here. Pretty much like Ian and John etc.