That's an Argument From Ignorance
Sometimes when atheists are presented with an argument for God's existence, they will say "Just because you don't know what could have caused X besides God doesn't mean that God did it!" This response, on its own, is weak. We all live our lives accepting beliefs on the basis that they are the best known explanations (i.e. because we don't know of a better alternative). Think about it: if you're called to jury duty and you have to examine a murder case, and all of the evidence seems to be utterly inexplicable unless the defendant is guilty, would you sit in the jury box and say, "Just because I can't think of a way to explain this mountain of evidence without invoking the defendant's guilt does not mean that he really is guilty!"
If a theist presents evidence of God, I think it could possibly be accepted as a basis for conversion, as long as it passes the following steps:
1. The evidence has to be something that is genuinely predicted by the god hypothesis or better explained if a God exists. Roy Abraham Varghese once pointed out that scientists don't know how bumblebees fly. Which is true, but how in the hell could God explain this phenomenon?
2. There should not be any competing explanations which are more plausible than the theistic explanation.
3. Given that theistic explanations have failed hundreds of times in the past, is this evidence currently being presented really so overpowering that I can ignore the lessons of history?
Jesus Mythicism as a Debate Tactic
There are accredited scholars, namely Richard Carrier and Robert Price, who defend the viewpoint that Jesus mythicism is somewhat more likely than the view that Jesus was an historical figure. There is a chance that they are right, and even that in the future more scholars will recognize their position. I think it's fine to look at their arguments and come to your own conclusion. The misconception that I have in mind here is not that they are necessarily wrong, but simply that their position ought not be used as a sort of tool for deconverting nonbelievers. After all, the vast majority of historians, including atheist historians like Gerd Ludemann and agnostics like Bart Ehrman, believe the best explanation of the facts is that there was an actual historical Jesus (Ehrman has written a book on the subject that is now available). The reason they believe this is because it is the most credible explanation they know for the evidence. Example: in Galatians 1:19 Paul says he met James, the brother of the Lord (Galatians is not forged, it's widely accepted as authentic). Since a mythical person cannot have a brother, and since this is from an authentic and very early source (written in the 50's AD) it's good evidence for a real historical Jesus.
I find myself constantly having to make a similar point to creationists who think they know geology, physics, and biology better than men and women (many of whom are Christians) who have spent their entire lives studying the issues. Needless to say, that is just silly. It may be OK to harbor private doubts and to raise questions about the issue, but no one ought to debate someone else under the pretense that they think they know the issue better than professionals. What's sauce for the creationists is sauce for the Jesus mythicists; Have your doubts and questions, but be humble and don't assume you know it all, and especially don't present a fringe viewpoint as a reason for someone else to accept your conclusion.
Since you can't prove a negative, the burden of proof is on the theist to show that God exists.
The statement "You can't prove a negative" is simply false. We can the President is not chinese, we can prove that one plus one does not equal three, I can prove that Julius Caesar is not my father, and so on. That being said, do atheists have the burden of proof? Yes and No. It is our responsibility to show that they have the burden of proof, but it is a fact that they have it. The principle of Ockam's razor says that all else equal the simplest explanation is most probably correct. If the world around us can be explained just fine without theorizing that a God exists, then we must judge it likely that God does not exist, because a simpler explanation (that nature is all that exists) is more likely than the more complicated explanation. A point often missed is that the argument from Ockham's razor does not merely argue for a lack of belief; it actively argues for the belief that God probably does not exist. That's why theists have the burden of proof: Because we've got an argument that shows God probably does not exist, and in order to get the probability above 50% again, they have to show that there is evidence that a God exists.
You can do even better than that. As I once put it: Atheism makes the distinct predictions that the only way complicated living things will exist is if evolution occured (because on atheism there's no person to create them, they can only be products of nature) and that the world will contain injustice and evil, simply because atheism means that there's no perfect being running the universe, and without such a person, it is overwhelmingly likely that bad things will happen from time to time. And that's exactly the universe we live in.
Atheistic philosophers have come up with a number of good arguments demonstrating the nonexistence of god, here are some handy resources on those:
Webpage: Evil Bible: The Impossibility of God (This is a brief and simple set of arguments)
Journal Article: Ted Drange, A Survey of Incompatible Properties Arguments
Webpage: Ten Atheistic Arguments
WebPage: Logical Arguments for Atheism
Webpage: Evidential Arguments for Atheism
Book: The Impossibility of God
Book: The Improbability of God
Richard Carrier Blogs: The God Impossible
A Common Argument Against Miracles
Thomas Paine made the following argument against miracles:
"Is it more probable that nature should go out of her course or that a man should tell a lie? We have never seen, in our time, nature go out of her course; but we have good reason to believe that millions of lies have been told in the same time; it is, therefore, at least millions to one that the reporter tells a lie."
Though this argument may sound airtight, it isn't. Let's look at a counter-example to see why: Suppose that some particular sequence of lottery numbers are only drawn one time in a million. Suppose you read that this number is the week's winning ticket in your local newspaper, which is known to give mistaken reports one time in ten thousand. Let us parody Paine: "Which is more likely, the mistake or the lottery drawing?" The manifest absurdity invites us to re-examine Hume's argument and correct the logic inherent within it. As it turns out, there is an error in this argument: it confuses how often an event happens with how often a report of that event is correct. In theory, it might be the case that miracles are exceedingly rare and that when miracles are reported the reports are usually true, in which case we would be justified in accepting a miracle based on mere testimony, since most of those testimonies would be correct.
Though this argument against miracles is flawed, I must point out that there is a correct argument against belief in miracles that has a similar result. I think a very heavy (though not impossible) burden of proof lies on any claimed miracle before it can be considered true. After all, investigations of alleged miracles have been undertaken in modern times, and they have turned empty, time after time. Little known fact is that the US Patent Office has officially refused to grant a patent to any claimed perpetual motion machine. No wonder. Every single claimed perpetual motion machine has turned out in the past to be invalid. Just a little bit of common sense tells us that, even if we can't absolutely rule out the possibility that such a machine might one day be built, it would take a fool to believe the next claim that turns up unless it was accomponied by extensive, exhaustive, extraordinary evidence. The same thing applies to miracle claims, for exactly the same reasons. And this doesn't forever and always end the debate over miracles. It could be the case that strong and overwhelming evidence turns up one day. But that has not, to my knowledge, ever happened or even come close to happening.
What's Sauce for the Goose is Sauce for the God
Theists will often present some fact that they think needs explaining: That a universe exists instead of nothing, that the universe is fine-tuned, etc. etc. A common atheist rebuttal to this is that if they get to arbitrarily suppose that God exists otu of necessity or for no reason at all, then why can't an atheist say the same about the universe? Likewise for the argument that the laws of nature come from God: if we can arbitrarily suppose that a lawfully thinking god exists for no reason, then why can't we arbitrarily suppose that a lawful universe exists for no reason? I used to go for this argument myself. It has gradually dawned on me that if this response is the only fallback response offered, the atheist position would end up being burdened with an awfully large number of arbitrary suppositions (that the universe exists as a brute fact, that it is suitable to life just by chance, and so on) whereas theism would only contain one supposition. Theism would be therefore simpler and, given that the simplest hypothesis is most probably correct, it would likely be true. Atheists shouldn't reach for such an intelluctually lazy response all of the time. Though this response might be valid if it was used in one and only one example, using it beyond that is problematic. And you don't have to. Little known fact is that there are valid and compelling nontheistic answers to every question like this that a theist might raise. Just look at Bertrand Russell's essay Why I am Not a Christian for an explanation about why there are laws of physics. Turn to Bede Rundall's book Why There is Something Rather than Nothing or Robert Nozick's Philosophical Explanations for discussions of why there is something rather than nothing. A quick and fun answer to this question is: there are infinite number of ways for there to be something (a universe composed of one atom, two atoms, three atoms... All the way up to infinity) and only one way for there to be nothing. The reason something exists rather than nothing is because something is more likely: It is infinity to one that something should exist.