This may or may not be connected with your Elizabeth.
My g-gf David Fairbairn was born in Innerwick East Lothian about 12 miles from Haddington and 5 from Dunbar. He was born 1859 and had an elder sister Elizabeth (b 1857). One generation back, his father Robert Fairbairn (must have been born c 1830-ish) was also from the area.
It's possible Robert had a sister who might have been your Elizabeth, which would account for naming his own daughter Elizabeth.
Not very convincing, but possibly worth bearing in mind.