Has her son obtained a copy of Frances’s birth cert as Scottish certs give both parents names plus mothers maiden name?
If the parents married in Scotland then their marriage cert will give their ages plus both sets of parents names plus mothers maiden names
If Frances was born 1934 - depending on when her parents married & their ages at that time - you may be able to find them on either the Scottish or Irish 1911 census
I am surprised that the birthyear & birthplace info on Frances in your opening post varies so much from her son’s info
