A possibility from an Ancestry family tree has an Elizabeth Bain, born 9 Aug 1824, in Ayr, Ayshire, and baptized 16 Aug 1824, in Ayr, Ayrshire (source given as: Catholic Baptismal record from St. Margaret's, and that it lists parents as Richard Bain & Jean Murphie. They have this Elizabeth Bain down as marrying John McGee, 21 Dec 1843, St. Quivox Church, Ayr, Ayrshire (source given as: Sarah McGee's 1865 birth register).
Linked to this tree is an Elizabeth McGie listed on the 1861 Scotland census in Muirkirk, Ayrshire, as coal miner's wife-but no husband listed, and with children William-11, Richard-8, Ellen-5, & Andrew-2. She is listed as born 1824 in Ayr, Ayrshire.