The new name is concomitant with a new covenant. The new covenant is very close to a restatement of the Lord's covenant to the Nephites (see 1 Nephi 4:14, 2 Nephi 1:20). Compare Lehi's statement of the promise to Benjamin's formulation of his covenant:
2 Ne. 1:20
20 And he hath said that: Inasmuch as ye shall keep my commandments ye shall prosper in the land; but inasmuch as ye will not keep my commandments ye shall be cut off from my presence.
Benjamin is proclaiming that although he is creating a new people "above" the old peoples who emigrated from Jerusalem, the promise of the Lord follows righteousness, and is therefore renewed in this new people. Thus it is a new people, but in religious covenant, and continuation of their mutual status as the people of God.