.
Edit: It's done more formally in the Catholic religion. IANAC, personally, so I'm unsure if it's still practiced regularly, but, the Rhodens, and HG, were not Catholic (at least I see no indication that they were).
The god father / god mother concept is still practiced regularly with Catholics. It may even be near mandatory for baptism. Though not all Catholics take the honor seriously, many do and maintain contact with their god child (children).
I am not sure either what religion the Rhoden family is. Though Catholics are not common in Appalachia, my guess is that there could be a relatively large number of them in south east Ohio. Then factor in the possibility of mixed marriages.