You have some amazing resources at your fingertips. I have found George Riddoch, his parents and his grandparents in Scotland. They were definately not from Ireland in Mary's generation.
I suspect that Mary Moren/Moran left Ireland either for work or because she was pregnant. Her first appearance that I've found is her living in Arborath, Scotland in the 1871 census with a 6 month old son. I found the son's birth cert and it says he was illigemate, no father named. She was listed as working in the mill, a very low paying hard job. Strangely enough, her future husband is listed as a boarder in her house , so perhaps he was actually the boy's father. His family was still living in town, so I would imagine he would have still had a place with them if he wished. The son, James Moren, is not present is any future census with the family. I couldn't find his death cert. All speculation.
While hard to read on the certs and spelled differently, I really think that Mary's mother's last name is something like Scandling, Scanlon, Scanlinn or however you would spell that name if you could barely read or write or spell.
I truely appreciate your efforts. This may be one that is never figured out.