What do we mean by "son of God?"
When Christians call Jesus "son of God" they using an epithet used of Messiah. It didn't literally mean biological son, but it did link the Messiah to God as uniquely related and as messenger of the divine. Messiah means "anointed" and is designated as one sent by God for a special task. Gentile Christians sort of took over that phrase and use it to designate Jesus overall relationship to God. While the specific doctrines of the Church state that Jesus is the incarnate logos, the second person of the Trinity. That is to say Jesus is "God come in the flesh." God became man and entered human history. This is essentially the Christian meaning of "son of God." It doesn't mean Jesus is literally God's son in the sense that I am my father's son, although that relationship can be metaphorically applied. The difference is that I am not equal to my father in authority in the family and I don't share the same essence with my father. He is a totally separate being than I am. So the term "son of God" when used by modern Chrsitians is a unique Christian notion that is historically based upon what was meant in the Gospels but has connotations the apostles never thought about. For this reason it must be used advisedly. I use it becuase it's part of the established language of the tradition.
Is there Proof he is the son of God?
In claiming Jesus is "son of God" we bring together several different layers of history and historical understanding of what that means. Included under the ruebrick of "son of God" would have to be the idea of Messiah, since the label began as a means of saying "this is the Messiah." We would have to discuss the creeds and what is meant in relation to the Trinity when we say that "Jesus is divine." This really means that the conventional notion of arguing that Jesus is the son of God is too simplistic. There's no way to give a clear cut argument when the concept itself is problematic.
Problem of Lewis's "Trilema."
Moreover, the traditional argument, C.S. Lewis's "trilema" is not that as clear cut as it seems. It's really a "neat" idea in that it's simple and to the point and is easy for the average person to get. He's either the Lord, a liar, or a Lunatic. He's too honest a great moral teacher to be a liar and too together and rational as a great moral teacher to be a lunatic. The problem is two fold: the argument implies that the whole issue of Jesus is based upon his moral teaching more than other things such as the nature of his birth. Secondly, there are other possibilities, it's not as easy as the argument makes it seem.
Rather than using that I go for a minimalist effort in arguing directly about "son of God." I merely argue that it's all tied together: his status as messiah confers upon him the title "son of God" with gentile updates focused through Trinitarian doctrine. I don't want to rest it all on just the Messiah stuff either because that would imply that the moral issues and the resurrection are not important. It all works together. In place of the Trilema I just argue that Jesus is in the running, he's the best candidate if we assumed what a candidate for Son of God would be like. He is a great moral teacher, his authority was recognized in his day in a unique way and it still is so recognized. His qualifications as Messiah tie in the two halves of Christian history, the Hebrew origins and the Gentle growth. The resurrection is the incing on the cake because it ties in to the Messiah concept (Messiah was to be the controller of life and would raise all of fallen Israel form the dead on the last day--Jesus is what Paul called "first fruits form the dead" meaning he's going first to show he has the authority to beat death--that's the kind of thing they thought about in that culture). What's being said in terms of "son of God" is both Messiah for the Jews and divine nature as God incarnate (incarnate logos) which Jews have never accepted and feel violates their basic concept of the sacred.
Sure there may be other possibilities to define Jesus but the option is clearly strong for understanding him in the traditional Christan sense.The arguments below set up the nature of Jesus as the best candidate to be son of God. These have to be taken in context with:
Messiahship
Primarily established by arguing for fulfillment of Biblical prophesy. Messiah was expected to be born in Bethlehem and Messiah was expected to control the powers of life and death, and so on.
Resurrection
Can be considered a proof in the context of Messiahship. Messiah was to control the powers of life and death and to raise all of fallen Israel on the last day of the world. That means Jesus own resurrection is a symbol as "first fruits" from the dead, "...Christ has been raised from the dead, the first fruits of those who have died" (I Cor 15: 20). This is how Paul understood the resurrection to point to Jesus' role as Messiah. Alfred Edersheim in his great work The Life and Times of Jesus The Messiah explains this as the meaning of this passage.
then:
Best Candidate for "Son of God:"
Ethical teaching in place, claimed himself that he was Messiah,
claimed divine nature, accepted as having great moral authority.
Greatness of Jesus Teachings and Character
Jesus Makes Unique Claims About Himself
Jesus' Moral Authority Recognized by Moral
Teachers Around the World.
These arguments are more like "soft sell." They are not strong arguments that independently prove Jesus' divine nature. They are more like prerequisites that have to be included to make the whole list of arguments complete. They are probabilistic arguments in that the best they can establish is that he has the better probability of being "son of God."
Existential encounter
Not to say that the only form of religious or mystical experience that has efficacy is Christian. In fact mystical experience around the world is universally the same in terms of the actual content of the experience and the specifics of each faith tradition seem like added on. Yet is also true that existential encounters specifically in terms of Jesus (calling upon his name, praying to him and so on) do work. They may be classified in the general category of this universal religious experience but they do also work when Jesus is specifically consulted. I can only answer for myself because I don't have other people's experiences. Yet the Christian tradition is full of examples of people with the same kind of experience. I am not saying Jesus is only one of many. My personal opinion is that Jesus is the reality and it is he who stands behind all faiths as the reality constant.I don't blame Buddhists if they see it in reverse. In my own experience I find this to be the case. I called upon Jesus and Jesus must be the one who answered because I keep calling upon him and he keeps answering. Empirical data does exist to support that claim, both in terms of mystical experience, and in terms of medical miracles.
(see also Lourdes)
Evidence on historical Jesus is not an argument for his sonship but underpins all study of Jesus.