“David08271977: In order to be from the Davidic line, a Jew must have Jewish paternity directly back to Dovid HaMelech. That would exclude any Jews with non-Jewish grandfathers.”
A son or a great grandson or “offspring” of King David cannot defined through modern genealogy. Documents and letters, which is the only real proof of lineage can only be traced back several hundred years. Everything else is obscured by time. For this reason, just carrying a name will not be suffice with oral tradition because adoption other ancestral issues. That is why I believe certain miracles will be proof enough that someone is the son of David.