Our listing of the sons of Mosiah numbers only four. Mosiah might have had other sons, but if he did this passage would be a little harder to explain.
It is a simple thing to pass over this declaration that his sons declined to accept the kingdom, but why would they do that? Certainly they would have all been trained from birth in their responsibilities to the kingdom. The one intervening circumstance is the spiritual transformation that the named four underwent. That experience does appear to have so transformative as to lead to the situation in which the four sons would abandon their inheritance for the opportunity to preach the gospel.
This would therefore presume that these four were the only sons of Mosiah, as any remaining son would have been hard pressed to find a reason to decline the opportunity to be a king.
It is quite clear from the dynastic histories of virtually all known kingships that competence would be a rare deciding factor in accepting the position. It is most likely that Mosiah had only the four sons, and that their decision to decline the kingship was a direct outcome of their complete change of heart.
Nevertheless, something else is going on that Mormon does not tell us.
Even if none of his sons would accept, lines of kingship are seldom so narrowly defined. In Mesoamerica such occasions were faced not infrequently in the king-lines among the later Aztecs, and the kingship would pass to a son of a brother, or perhaps to a son of a son.
Not only must his sons have declined, but there must not have been any other possible heir for this coming change in government to occur because of a rupture in succession. While it is possible that there were no other possible heirs, it is equally possible that Mormon is skipping over some of the details and simply going to the heart of the issue, which is the transition from kings to judges among the Nephites.