We pray in Jesus name because he is the reason that we have access to God. Without the atonement, we would be unreconcilably impure, and therefore excluded from God’s presence, for “there cannot any unclean thing enter into the kingdom of God.” (1 Nephi 15:34.) It is only through the Atonement that our uncleanness is made clean, and therefore only through his name, as representation of his atonement, that we may approach God in prayer.
As the mediator between us and God, Jesus opens the door to our ability to pray to the Father. In this case as the twelve and the multitude pray to him, he redirects their prayers to the Father. When Jesus says that: “I pray unto thee for them,” he is redirecting their efforts. They are praying, and the only one able to answer those prayers is the father. Therefore Jesus prays for them directly. In this instance he literally becomes the mediator that his atonement is for all of our prayers.
The prayer that the people offer, and that Jesus opens to all who accept the gospel, is to become “in them as thou, Father, art in me, that we may be one.” This language echoes Jesus’ teaching in the New Testament:
John 17:20-22
20 Neither pray I for these alone, but for them also which shall believe on me through their word;
21 That they all may be one; as thou, Father, art in me, and I in thee, that they also may be one in us: that the world may believe that thou hast sent me.
22 And the glory which thou gavest me I have given them; that they may be one, even as we are one:
As in the Old World, so in the New. Jesus prays that those who believe will become one with the Father, as is Jesus. What is this “oneness?” It clearly cannot mean a corporeal unity, but must be indicative of a conceptual unity. It is a recognition that while there may be diverse people, we can all be of “one heart.” (see 1 Chronicles 12:38; 2 Chronicles 30:12; Jeremiah 32:39; Ezekiel 11:19; Acts 4:32; 2 Nephi 1:21).