Benjamin is making another transition here. The first transition is from “blood” to “name.” In previous verses, the “blood” was significant. Now he has changed to emphasize the “name.” This is critical to the end of his discourse, which is to give his people a new name as symbol of their unified covenant. With this shift, Benjamin is using the ancient conceptions of the power of the name to his benefit. Modern man understands that it is the person rather than the name that is important, but for the ancient world, the name in and of itself held power. Modern English retains some of this conception in the phrase “speak of the devil.” The original context of that phrase was that one could invoke the presence of the devil by speaking his name – thus “speak of the devil, and there he is” (this last part is sometimes used on the phrase, but often left to be implied in modern usage).
The next important transition moves from children to all men. Benjamin accomplishes this transition deftly by having salvation come “unto the children of men.” This phrase both combines the idea of children which ties to the previous verse, and the metaphorical meaning that allows the phrase to encompass all humanity (as all are “children” of someone).
Internal Reference: The idea of the name of Christ being of salvific power was also taught by Nephi: “2 Ne. 31:21 And now, behold, my beloved brethren, this is the way; and there is none other way nor name given under heaven whereby man can be saved in the kingdom of God. And now, behold, this is the doctrine of Christ, and the only and true doctrine of the Father, and of the Son, and of the Holy Ghost, which is one God, without end. Amen.”