Patronymics were - maybe still are - in common use in the fishing communities of north-east Scotland, where there seemed to be an unwillingness to extend the available name-list, as it were.
In the era I'm thinking of, there just were no Waynes, Shanes, Brads and so forth. Thus, calling someone Jeems's Willum (James's William) was a method of distinguishing that man from Ailick's Willum (Alex's William).
I don't believe there actually is a word to cover the circumstances you describe, Coco, but if you do discover such a thing, I'd be glad to hear what it is.