Background: My parents were having a discussion of how name suffixes can be utilized if I started having children, how would name suffixes occur. This was all hypothetical.
My father, and let's call him, Bob, claims that if I (who has a different name from Bob) have a child named, "Bob", then I will be able to name my child Bob II (Bob, the second) or Bob Jr. He thinks that regardless of a generational gap between the child and father, his grandson is allowed to have a suffix for his own name that indicates a succeeding suffix for the name of Bob.
My mother claims that I would not be able to name my child Bob II or Bob Jr. because in order to have such suffix requires a continual generational naming convention. For instance, if my name was Bob, then I can name my child Bob II or Bob Jr. if and only if my name was Bob. Therefore, if my father's and child's name was Bob, it has to be Bob and only Bob without any suffixes.
I've tried looking for laws in the United States that says anything about suffixes for names but I haven't found anything.
Question: Are there any laws in the United States or any of its states that restrict the usage of name suffixes, such as II, or Jr., or IV, etc.? Whose claim was correct about when it is legal to use name suffixes?