Funny, I saw this name and had to respond. Your John C Mayhew lived with my 2nd great grandfather, Jas Ed Brown, in 1880. I'm assuming he was a farm laborer.
If you have the date of death, which you do, it may be that you can peruse the old Rke Times a day or two after that date, but he would likely not have had an obit there, if he were in Franklin County.
My advice would be to contact the Franklin County Courthouse or the Franklin County Historical Society...to ask if there were any newspapers from Franklin County...that would be available somewhere to search through. I've looked for old Rocky Mt papers and had no luck, myself, but that is the best advise I can offer. I do hope you the best of luck.