A dominating theme of the Book of Mormon is the restoration of Israel. In our day, the House of Israel are those who believe in Jesus Christ and in the true doctrine.
6 For thus it behooveth the Father that it should come forth from the Gentiles, that he may show forth his power unto the Gentiles, for this cause that the Gentiles, if they will not harden their hearts, that they may repent and come unto me and be baptized in my name and know of the true points of my doctrine, that they may be numbered among my people, O house of Israel;
22 But if they will repent and hearken unto my words, and harden not their hearts, I will establish my church among them, and they shall come in unto the covenant and be numbered among this the remnant of Jacob, unto whom I have given this land for their inheritance;
As we repent and hearken to the words of Christ, we become part of the Abrahamic covenant and regardless of actual lineage are part of the House of Israel. Believers become heirs to the promises given to Abraham and his descendants and also assume the obligation given to Abraham, Isaac, and Jacob that through them and their children all the nations of the earth would be blessed.