I can only speak from my personal experience here, but most of what has convinced me -- has not been the evidence (which thankfully, is not absent), but the texts themselves which carry too deep a sincerity to be missed.
I might not be a professional historian, but I am fast becoming a very enthusiastic layman -- I remember when Ehrman chatted with Witherington about his book on this subject, and what emerged was this same sense of the deep sincerity of the writers. I could go to include so many great minds who have had this same experience with the texts. It's what historians look for, and most of them find the NT has a lot of that. It makes for compelling reading, but oddly there seems to be the other something about it -- and that is it holds such a high standard of sincerity that it almost become too much to manage at times. Do you know what I mean?