Before beginning, I should point out that I'm an atheist. I don't consider the existence of god(s) impossible, but I do consider the existence of a god about as improbable as the existence of gnomes or unicorns. Second, I am going to be using the word 'God' a lot in this post. I will not just be referring to the Judeochristian God, though, but to any and all gods.
A while back I got into an interesting debate with a Christian acquaintance of mine, during which I asked him what kind of evidence could convince him that God doesn't exist (or at least that his existence is very unlikely), to which he, like most other theists when asked a similar question, replied that nothing could convince him of that, as he had faith. Realizing that his belief was dogmatic and entirely unfalsifiable and that debating him probably wouldn't be of any use, I quickly ended the debate. I have wondered, though, if these are not hypocritical thoughts. Yes, if solid evidence for God's existence would turn up, I'd start believing in him, but I have never really thought about what I would consider solid enough evidence for God's existence. So I have been thinking about it now, and come up with several possibilities for evidence that could convince me I'm wrong:
First, a message from God himself would make me consider his existence very likely. I'm not talking about 3000 year old books or a bread in Wherethehelleveristan having spots that vaguely look like Jesus' face, though: I'm talking about the stars moving to form the phrase "I am the Lord thy God, thou shalt have no other gods before me." (or a similar message. The actual content, I suppose, wouldn't really be important). If the stars actually did that, I'd be almost completely convinced of God's existence. Of course, it could be the work of aliens, but to move all those stars or create a highly complicated gravity-lensing effect, they would have to be a Type III civilisation on Kardashev's scale (I'm afraid I don't have time to talk about Kardashev's scale now, but it's an interesting topic and I will probably post about it in the future), and I think we would have detected such a civilisation anywhere near us, as they would leave incredibly large traces of their existence on at least a whole galaxy. Besides, what could possibly be their motivation for such an act? Maybe they think we might become a problem in the future, so they're trying to slow down science by sending us into a religious dark age, but if they are, why not just wipe us out? They would have the power to move stars, after all.
A smaller version of the moving stars idea, like gigantic letters made of clouds in the sky or a gigantic voice out of nowhere saying a message, would work too, for me, though less convincingly so. It would be less convincing, as the technology needed to do such a thing would be much lower, and thus aliens doing it would be a lot more likely. Still, it does seem highly unlikely that aliens would bother doing something as weird as that, so I'd say God would still be a likelier source of such a message.
I would not be convinced, though, by any message that only I could percieve. If I start hearing a voice in my head that tells me it's God, Occam's Razor says that the most likely explanation is that I've gone delusional, and I'd visit a psychiatrist, not a church.
A second thing that would make me consider God's existence a lot more likely would be a miracle. The miracle would have to be well-documented, repeatable, and completely unexplainable by science, though. And of course, if a natural explanation for the miracle was ever found afterwards, I'd reverse my judgement on the matter.
An accurate prophecy would also make me consider God's existence very likely. It would have to clearly refer to an event that was in the future when the text was written, though, preferably containing exact names and years. If the Bible contained a phrase like "In the year 2667 ad Urbe condita, the Archduke of a land called Austria-Hungary, which will be situated in Dacia, will be assassinated by a Gavrilo Prinzip, and this will lead to war between Germania, Britannica, Gallia, and Scythia.", I would consider this a very compelling piece of evidence. I would not be convinced, though, by vagaries that could refer to World War II, 9/11, the Chernobyl catastrophe, or your mother's awful cooking depending on your interpretation.
This one kind of fits in with the prophecies: if holy books contained accurate scientific knowledge that hadn't been discovered yet at the time, I would consider this strong evidence for God's existence. If in a holy book it would clearly and unambiguously say that the Earth orbits around the Sun, which moves around the centre of the Milky Way, or if it accuratly predicted the properties of Uranium and identified it as having 95 protons and 144 neutrons long before it was even discovered, this would be very strong evidence indeed. Or if it contained advanced medical knowledge about penicillin and vaccins (Why indeed didn't Jesus give his followers knowledge about those and lighten the burden of 18 centuries of diseased people?), or something like that.
Another thing that would convince me of God's existence would be if one particular religious groups significantly lived longer, healthier, and happier lives than everyone else and won all their holy wars, as long as this is not explainable by any other factors. Prayer from this group actually being clinically proven to improve a diseased person's health would also help convince me.
Now, if God actually existed, most of these evidences would be pretty easy for him. I admit the stars thing may be a bit too big (after all, he'd have to disturb a whole lot of aliens by moving their homeworlds several lightyears), but letters in the clouds wouldn't be much trouble for a guy who created the universe, and it would get him loads of believers. If the Bible is to be believed, miracles used to happen all the time, so why can't we ever get one confirmed, especially considering that doing miracles is pretty much what God's job is all about? If he is omniscient, why are all his prophecies so incredibly vague, and why didn't he give humanity some accurate science and medicine? And if he actually existed and answered prayers, wouldn't my last possibility for evidence arise automatically?
Yet none of these pieces of evidence for god(s) are ever found. The reason why is simple, and it's the same reason that no biologist studying mushrooms has ever found a gnome living inside one: because gnomes don't exist, and neither does God.