Richard
From what I can see John Archibald Stuart Paton was born in Tillicoultry in 1872 to his father John. He had at least 4 older brothers, none of whom were called Robert. He also had a younger brother and he wasn't called Robert either. At some point before 1901 census he moved to Hook Heath, Woking and appears to have stayed in the area until his death in 1946.
He appears to have remained single at least up to 1939. I suppose it is possible that he had a son or that some other relative was called Robert but it certainly doesn't appear that he had a son, brother or father were called Robert. At the end of the day the book sold a reasonable number of copies and Paton isn't that uncommon a name so perhaps a family connection to Stuart Paton is unlikely.
Niall
Well found Niall
According to Ancestry, Paton's father John had a brother named Robert, born 1845. Unfortunately said brother died as a young boy in 1850.
Rich, if you have an Ancestry subscription, find Paton using the info provided by Niall and dig around -- see what you can find.
Edited to add: I found evidence of Paton playing a match for the Oxford and Cambridge Golfing Society so he must have attended one or other university and won his Blue. But I haven't been able to find any more evidence yet -- another route of research.
Edited a second time to add: When Paton died in 1946, according to the Probate Register, he left an estate worth £73,000, which translates to close to three million in today's money. His father is listed in the 1881 Census as 'junior partner of John Paton & Co Tilicoultry'. Trying to trace the company.
Edited a third time to add: I'm pretty sure they were textile (woollen) manufacturers.