Hi
Civil Registration did not commence in Scotland until 1855 so if she was born earlier then you are looking for a church record of the baptism. Unfortunately not all denominations records are online, Familysearch has indexed many of the registers for the Church of Scotland, and a small number of others, but they have few Free Church of Scotland, Episcopal, United Presbyterian etc records. Scotlandpeople has online the OPRs, which are Church of Scotland, and Catholic Parish Registers but not other denominations.
As far as the surname is concerned the baptism may be under the mothers name only but you would need to find the baptism to know for sure, it was not uncommon for children, even after Civil Registration commenced, to adopt the surname of the reputed father. Sometimes an illigitimate birth would be registered by the mother with no father named but with the reputed fathers surname as a middle name. The child would then drop the mothers surname and simply use the reputed fathers surname (the childs middle name) as their surname.
Andy