The claim of God is basically a claim of a being that exists that can do anything.
With such an extreme claim you shouldn't be surprised that the evidence required for such a claim would also be fairly difficult.
The theist likes to pretend that the atheist is just intransigent because the claims they make are very difficult to evidence and often nebulously defined in an unfalsifiable manner.
I'll second variant on this: it makes sense that we would need greater proof for a being like god's existence than we would for the existence of a previously-unknown animal species....or really, just about anything else. The existence of an immaterial god who is omnipotent is quite a grand claim, whereas the existence of a new animal species really isn't contrary to anything in our everyday experience.
Wow, there were a lot of posts in three hours. I want to pick up on these two responses to mine, and in doing so also make reference to the points on falsifiability and burden of proof.
Although it is being claimed that claims made for belief in God are unfalsifiable, the atheist position is equally unfalsifiable. For most of their proponents on both sides, these beliefs are ultimately faith positions that are not falsifiable (ie. there is no evidence that could ever be put forward to make the person change their mind).
Falsifiability is about setting out reasonable criteria that give us the best chance of arriving at the right conclusion - ie. in this case, that if God exists then our line of argument will probably lead us to conclude that he does, and that if he does not then our line of argument will probably lead us to conclude that he does not. It is very important not to set a burden of proof so high that, in one direction or the other, it can never reasonably be met.
It is also important to consider all the evidence, and not only the strands of it that suit our viewpoint. For example, it is relevant evidence that a very large proportion of people through the ages and still today, including many who are phenomenally intelligent, have believed in God. This sets belief in God apart from belief in fairies or the Great Pumpkin. It is false reasoning to claim an equivalence between these different beliefs.
It is also unreasonable to make demands of God that he should not be expected to make, which I think is where Variant's first point was headed. I had a friend at university who said, "If God will come and sit on my bed and talk to me in person, then I'll believe in him." To which I responded, "Why should God do that? It's you who need him, not he who needs you." My friend was trying to make God into this kind of genie in a lamp who jumps at our beck and call; but such a being is not the God of the Bible, and therefore to require this of God in order to believe is not reasonable. It would be like me saying, "No, I'm just not going to obey the law unless the Prime Minister comes and explains it to me in person." Why should he do that? That isn't what the Prime Minister does or should be expected to do.
I think most of the philosophical and seemingly scientific arguments that try to prove or disprove God don't lead very far. The design arguments that have been discussed a little on this thread are ultimately unconvincing, because they depend too much on certain knowledge of things we just don't know, so they are only really convincing to people who are already theists. (That's not to say I think that the argument that it all came about by chance is convincing - just that there are too many unknowns to be able to say with any certainty that it didn't.) The arguments about why God allows suffering in the world that are the main argument put forward against belief in God are also unconvincing; there are many ways that these questions can be answered if you are so inclined that are perfectly logical; but equally, a person who is minded to disbelieve can easily persuade themselves that the questions have not been answered.
However, I do think the evidence around the resurrection of Jesus is very much worth dealing carefully with. It is the closest we can get to a falsifiability criterion. Paul writes that if the resurrection didn't happen then Christian faith is worthless (1 Cor 15:17). On the other hand, if it did happen, it is a very serious blow for atheism. I would recommend anyone who is genuinely interested in truth to examine this closely.
Again, we need to be careful not to set the bar for acceptance or rejection so high that we can never be convinced. We wouldn't do that with other beliefs, so why with this one. A lot of Christian arguments often come down to "The Bible says so so it must be true", which is not a falsifiable position. However, atheist arguments seek to turn every small area for doubt into a major challenge, which is also disingenuous.
I am not going to wade into all the arguments about who first arrived at the tomb on the Sunday morning and about whether the angels appeared to the women before or after Peter got there. That is the kind of detail we can't expect nor do we need historical certainty on. Neither is it reasonable to exclude the Bible or other Christian witnesses from consideration; it is hardly surprising that all the people who believe in the resurrection are Christians: if you become convinced of it then of course you will become a Christian. And the Bible, even if you regard it as no better than other historical sources, should also not be regarded as worse than other historical sources. In fact, secular historians generally rate its reliability pretty highly. Note also that, as a historical source, it is not a single book but is a collection of books from different authors as well. So it is not just one witness's account, but several.
The detailed questions of exactly what happened on that Sunday morning are largely unimportant for our study. However, the broader question of how early Christianity got started at all unless Jesus actually rose from the dead is not dependent on any one account or detail. There are strong traditions that all the disciples except possibly John were persecuted and killed for their claim that they had seen Jesus after he had risen from the dead. That makes their testimony pretty credible, because they were not dying just for their unsubstantiated belief in something, as modern Christians or Muslims or people of any faith persuasion might do, but for things that they had personally seen, and that if they were not true then they knew they were not true. Generally, people may die for things they believe in, but they don't die for things they know to be false.
Paul, writing probably less than 20 years after the events in question, writes of things that were passed on to him as tradition (and therefore have a very early origin), and he also writes of more than 500 witnesses who saw Jesus after he had risen, most of whom were still alive at the time of writing. Now, if he were writing this in a book that was being published to convince a sceptical audience then we might have reason to question his motive in writing it. But he is writing to people who already believe, mentioning the fact almost in passing as a clarification of a point that his hearers had misunderstood. There is no reason to suppose he expected this writing to be published any wider than to the church he was writing to. So his claim here is worthy taking seriously.
Of the various explanations for how the story of Jesus' resurrection got started, the one that is far and away the most plausible is that he actually rose from the dead. Claims of a stolen body raise the question of who. And in any case, if Jesus remained dead, whoever took the body, why would the disciples die for their claim that they had seen him risen? Claims that Jesus didn't really die are pretty implausible; even more implausible are the claims that he never even lived. Claims of hallucinations might be plausible if just one or two people had them, but if so many did, and were sufficiently convincing to be able to get people to risk social ostracism, persecution and even death on the basis of their claims, then this is stretching the boundaries of plausibility to past breaking point.
The resurrection accounts, then, I submit, provide our best candidates for falsifiable evidence, that really deserve close scrutiny. For anyone who genuinely wants to claim that they have carefully considered the question of God's existence and rejected it, they have to come up with a credible account of the rise of early Christianity aside from Jesus actually having risen from the dead. If they don't want to do this, their disbelief is every bit as much an unsubstantiated, unfalsifiable, irrational position as they claim of any religious viewpoint.
Roonwit