In the comments on Mosiah 3:17, the concept of the bonding between the name and the essence of the person was discussed. It becomes relevant here because the name of the Messiah is now applied to a people covenanting with and through that Messiah. Benjamin declares that his people shall be known with a new name, and that implies a new beginning, and a new character infused with the principles Benjamin outlined in his speech.
Why does Benjamin rename his people? The change in the name is intended to provide a change in identity. There were people of Nephi and people of Zarahemla. It would hardly be surprising to find that much of the internal division was along lines following the previous religious and political understandings of those two separate peoples. Benjamin is anxious for a new beginning after that terrible civil disruption. He attempts to change the discourse from separate Nephites and Zarahemlites to a single new people where all follow, and are named, for the coming Messiah.
Benjamin desires that they “should remember to retain the name written always in your hearts.” It is an ambitious political move and a very optimistic one. Unfortunately, Benjamin’s desire that they always remember their new identity as one people dissolves during his son’s reign.