The father of Lamoni confers the kingdom upon a son and renames his son. It is not unusual in many cultures for a king to take a regnal name upon assuming the throne, and indeed this very practice can be documented for the Classic Maya based on the evidence of the glyphic texts. (Simon Martin and Nikolai Grube. Chronicle of the Maya Kings and Queens. Thames & Hudson, 2000, pp. 81, 91).
What is mildly interesting in this case is that the king adopts the name of the people rather than the people and land adopting the name of the king, as had been the case in the past. Name changes for peoples in the Book of Mormon tend to occur when there is a new dynasty head in a new location.
In this case we have a combination of events coming together. There is the clear attempt at a designation of a new people because of the name of Anti-Nephi-Lehi. It would appear that the designation of the regnal name as the same name as will be adopted by the people is intended to create a further separation from the old and an establishment of the new. Nevertheless, there is no real indication that the people of Anti-Nephi-Lehi have yet removed themselves from their previous cities. In fact, it appears that they have not. Thus we have a situation where there is a mass conversion in multiple cities that we must assume did not affect every member of the city. In spite of the lack of complete conversion, we have the creation of a new polity that must attempt to incorporate all, whether converted or not.
What we appear to have, in the context of the Mesoamerican locale in which these events are suggested, is a conversion of ruling clans. The clans likely have some intermarriage relationships, for we know that Nephi and Ishmael were ruled by father and son. It is probable that such clan relationships may have existed in at least some of the other cities. The conversion of the ruling clan would establish the alteration of the ruling structure for the entire city, but still allow for the dissention of other clans. Thus we can have the rather complete change of governance that we see, and still have the apparently smaller body that eventually arrives in the land of Zarahemla.