Mosiah 1

As King Benjamin instructed his son Mosiah to gather the people at the temple that he might speak to them, he shares one of his motives -

"...I shall give this people a name, that thereby they may be distinguished above all the people which the Lord God hath brought out of the land of Jerusalem; and this I do because they have been a diligent people in keeping the commandments of the Lord.
12 And I give unto them a name that never shall be blotted out, except it be through transgression." (Book of Mormon | Mosiah 1:11 - 12)

We see in Old Testament times, Abram, Sarai, and Jacob (Abraham, Sarah, and Israel) all received new names as they took upon them sacred covenants and promises. Saul later took upon him the name of Paul. Isaiah said the same of those that receive celestial glory that they -

"...shalt be called by a new name..." (Old Testament | Isaiah 62:2)

And to him that overcometh or keeps the covenants and commandments of the Lord, John writes -

"...will I give to eat of the hidden manna, and will give him a white stone, and in the stone a new name written, which no man knoweth saving he that receiveth it." (New Testament | Revelation 2:17)

But this name that would be given to his people if they were faithful had more to do, which in my mind is a last name, much like the name we take on us acknowledging our parents, was as King Benjamin would teach his people was Christ for -

"...there shall be no other name given nor any other way nor means whereby salvation can come unto the children of men, only in and through the name of Christ, the Lord Omnipotent." (Book of Mormon | Mosiah 3:17)

Much as we did at our birth in the mortal existance, King Benjamin taught -
"...now, because of the covenant which ye have made ye shall be called the children of Christ, his sons, and his daughters; for behold, this day he hath spiritually begotten you; for ye say that your hearts are changed through faith on his name; therefore, ye are born of him and have become his sons and his daughters." (Book of Mormon | Mosiah 5:7)

As we make and keep sacred covenants, we receive new names or covenant names. Not only a new name but also the name of Christ as we are born of Him. Thus as we renew our covenants weekly, we covenant that we -

"...are willing to take upon them the name of thy Son..." (Book of Mormon | Moroni 4:3)