Mosiah 18

At the Waters of Mormon, Alma teaches his followers what it means to take upon oneself the name of Christ saying -

"...as ye are desirous to come into the fold of God, and to be called his people..."  (Book of Mormon | Mosiah 18:8)

Being called "His people" suggests taking upon themselves the name of Christ...being -

"...willing to bear one another's burdens, that they may be light;
9 Yea, and are willing to mourn with those that mourn; yea, and comfort those that stand in need of comfort, and to stand as witnesses of God at all times and in all things, and in all places that ye may be in, even until death..."  (Book of Mormon | Mosiah 18:8 - 9)

Being baptized -

"...as a testimony that ye have entered into a covenant to serve him until you are dead..."  (Book of Mormon | Mosiah 18:13)

And as priests and saints, they -

"...were not to depend upon the people for their support; but for their labor they were to receive the grace of God, that they might wax strong in the Spirit, having the knowledge of God, that they might teach with power and authority from God."  (Book of Mormon | Mosiah 18:26)

In my mind, taking upon the name of Christ mans to serve him for His grace.