Now Jesus did many other signs in the presence of the disciples, which are not written in this book; but these are written so that you may believe that Jesus is the Christ, the Son of God, and that by believing you may have life in his name.
Which would seem to indicate that John's intention is to answer this question: Who is Jesus? The answer he provides: The Christ (or the Jewish Messiah, the anointed one from God), the Son of God.
However, in an article published in the Journal of Biblical Literature in 1987 (and then more briefly in his excellent commentary, the Gospel According to John), D.A. Carson makes the argument that we ought to understand those words in a slightly different order. His argument essentially boils down to this: in v31, the Greek phrase, ἵνα πιστεύητε ὅτι, translated so that you may believe that refers not to who Jesus is, but to who the Messiah, the Son of God (terms he takes basically as synonyms) is. So the question then becomes: Who is the Messiah, the Son of God? Answer: Jesus.
This makes a lot of intuitive sense to me as I read John's account of the gospel. I'm not by any means versed enough in Greek to make much of a judgement on Carson's linguistic argument, though I sat with it long enough yesterday that I'm pretty sure I grasp what he is saying. To me the question is, which of these questions makes the best sense of the book?
In the end, either question is going to give us substantially the same answers, because John is going to labor who Jesus is, where he comes from, and what his mission is. But the additional information he provides, the nature of his selectivity in constructing the narrative of signs (see v30), these will be shaped by the foundational nature of which question he is asking. As I'm prepping to preach this text on Sunday, I'm convinced that Carson is correct. The question driving John is, Who is the Messiah? As John records the statements Jesus made, the signs he performed, his substitutionary and atoning death on the cross followed by the Resurrection, it all points to this world transforming answer: The Messiah, the Son of God, the long awaited one is none other than Jesus of Nazareth.
To an audience of unbelieving Jews (again argued for by Carson based not just on his understanding of this text, but of numerous factors throughout the book), this would be compelling evidence to come to Christ. He is the one they were waiting for. And in believing in him, life is gained. Life with Messiah. Life with God.