Now, I am really not trying to be snarky here, so please hear me out. Try to honestly ask yourself the question:
what evidence would make you reasonably sure that a UFO claim is in fact an alien spacecraft?
Would it take 1 photo or 100? Would it take 4 testimonials from eyewitnesses, or 50? Would it depend who reports it, like the New York Times or NASA? Would you need someone to stake their reputation on it, or their life? Would you require physical evidence, or would a report suffice? What would it take? If you really wrap your head around this question, and come up with a possible list of evidence that would possibly sway your opinion, as well as the reasons why that evidence would be convincing, you’ll start to understand what an atheist thinks of the resurrection of Jesus.
When the day is done, the only evidence for the resurrection comes from (at best) four (so-called) eyewitness testimonials written decades after the events and the possible (although extremely dubious) reports of a handful of people believing so strongly they they laid their life down for it. There is no physical evidence, and no more than a handful of extra-biblical references (again, with dubious or weak content). From all of the arguments I’ve heard, this seems convincing to Christians.
Compared to UFO reports, for every eye-witness claimed for the resurrection, I can produce 100 witnesses for alien spacecraft – with direct reports within hours of the events. For every supposed apostle who is claimed to have been martyred, I can produce several people giving their lives for the belief in alien spacecraft (think, Heaven’s Gate here). For every extra-biblical reference, I can give you thousands of photos, metal implants and alien abduction therapy session reports – all of which is stronger evidence than any of the extra-biblical references to Jesus (some of which are forged). Yet, despite all of this, I am not at all convinced that UFOs are alien spacecraft, and neither are most people.
Why is it then, that people are convinced by the –far less– evidence for the resurrection? What’s the difference?
Professor Brian Blais' Blog 
Have you read “The Believing Brain” by Michael Shermer? He talks about how belief comes before reasons for belief. “Belief dependent realism”, I think he calls it.
People don’t believe in the resurrection because as unbelievers or without any prior opinion they came across compelling evidence for it, and became convinced. They already believed it (“by faith”, as they say), and if you ask for evidence they’ll show you what they have, but it’s beside the point.
Belief in UFOs, on the other hand, is not a belief system with the deep (both historically and culturally) underpinnings of Christianity. Children are not sent to Sunday-morning classes where they are taught UFO lore starting at early preschool age. There is not an entire culture all around them that reinforces belief in UFOs at every turn.
But to answer your question: it would not take much. I remember a TV show (can’t remember which one it was — one of the ones that briefly appeared in “The X-Files” wake) where one person was telling another of an alien encounter he’d had. The latter was skeptical until the first person took him to his home and showed him an artifact that had been given to him by the aliens. It was a very small thing — about the size of a credit card, I think, but it sort of expanded in a kind of 4-dimensional fashion and hovered in the air, spinning and undulating (presumably in 4-dimensional space). That was all that was needed to convince the skeptic.
The key is, “extraordinary claims require extraordinary evidence”.
The thing is, the God described in the Bible seemed to understand this. On many occasions (4 or 5 I think) Jesus is recorded as saying something along the lines of “these miracles I’m doing are to provide evidence that the things I’m saying are true”. And the Apostle Paul says something similar a few times in the New Testament. And remember the story of Elijah calling down fire from heaven to consume an altar saturated in water? Elijah in that story might as well have said “extraordinary claims require extraordinary evidence” — that was the whole point of soaking the wood in water, and filling a trench around the altar with water.
Of course, all we have now are the stories of these events recorded in ancient documents, and that’s the problem. Stories describing spectacular events that served as the “extraordinary evidence” for people 2,000 or more years ago is not “extraordinary evidence” any more. Had I been there to, like Thomas (my favorite disciple), witness Jesus’ death and burial, and then see him alive again, and “thrust my hand into his side” — well, that would have sufficed for me. Jesus said, somewhat derisively, but accurately nonetheless, “unless you see miraculous signs and wonders, you will never believe”, and he proceeded to give the people what they needed. He just doesn’t seem interested in convincing anyone any more, in that way.
I think Michael Shermer (I’m reading “The Believing Brain” right now, so it’s on my mind) asked himself the question, “what sort of evidence would it take to convince me of the existence of God?”, and his answer was something like “nothing could convince me, because no matter what kind of experience I had — no matter what wonders I saw, it would always be far more likely that the whole thing was a hallucination or some other kind of cognitive misfiring.”
Yet unlikely things happen. There are billions of people in the world, so sometimes billion-to-one things happen, and we usually don’t see them happen. Sometimes we’re told of an unlikely event, and it’s the truth.
So, what sort of evidence would convince you of, let’s say, the resurrection of Jesus?
Very good question. I’m travelling, but plan on answering this as a full post, because it does require comment.