See if I ever again leave you "free to focus on other matters!" That being said, you raise a good question with regard to the Christian teaching of forgiveness through Jesus. I do not pretend to be a scholar on the subject (either the particular example you raise, or the larger matter of Christology). Still, I harbor strong impressions, which I hope are based primarily in reflection on biblical texts, Christian history and experience.
Christians view Jesus through the lens of the resurrection. All his words and actions take on additional meaning for us as a result. To the best of my knowledge, all early Christians interpreted the resurrection as God's validation of the life and teachings of Jesus. Almost immediately, they concluded Jesus must be the Christ. Some tended to view him as a man, whom God elevated to the status. Others soon began to see Jesus as the incarnation of the Word (to borrow John's language). We see a bit of both tendencies in Paul's letters. Ultimately, the Christian doctrine of the Trinity evolved from such considerations. With regard to your question, Christians felt it natural to pray in the name of such a Jesus or to ask forgiveness in his name.
The practice also derived from early Christian reflection on other sayings of Jesus. For example, Jesus told his disciples that whatsoever they asked in his name would be granted, that no one came to the Father but through him, and the like. You and I live in a time when sets of scholars debate which sayings should be regarded as authentic. The first followers seem to have accepted and worked with whatever sayings were available. That being the case, they soon began to pray in his name and seek forgiveness through him.
Experience played a role as well, I think. The first generation of Christians clearly believed Holy Spirit was active in and among them to comfort, bring to memory the teachings of Jesus, guide them in ministry, and instruct them. My personal hunch is that if we could interview early Christian leaders, they would tell us they believed Holy Spirit led them to pray in the name of Jesus and seek forgiveness through him.
All of the above begs the question of Jesus' intentions. How one deals with the matter seems largely determined by whether one factors genuine resurrection into the equation. By that, I mean we're either dealing with a life and its attendant perceptions/intentions which ended with death, or with someone who lived bound within the limitations of genuine incarnation then lived again with expanded perceptions/intentions via resurrection. The early Christians went with the second option.