So I'm not one to be a fan of logical arguments for the existence of God. I think I stand with Kierkegaard in thinking that God's God-ness is an objective truth which we access subjectively. However, I was thinking about this last night when I was trying to fall asleep. I don't see this argument as so much a "therefore, God exists, QED" as much as a "therefore, it's reasonable to entertain the possibility."
So it seems like the argument against God's existence tends to include one general premise: we have no evidence for the existence of God, and (if we take "God" as the God written about in the Bible) we in fact have empirical evidence against His existence. There seems to be an implied premise which goes along with this. Things as they really are must correspond with the way we experience our, perhaps relative, reality. This premise, I think, is something to dispute.
Here's my argument: If God (in the way God is characterized in the Bible) exists, then we must come to know God through faith. Faith, by definition, requires one to believe something which does not correspond with that person's relative reality. Would it not make sense, then, if one must come to know God through faith, that He would create us to experience reality differently than He does? If this is the case, then the argument that God can't exist because His existence is inconsistent with the way we experience reality doesn't stand.
This argument can't be a formal argument because it leads to the fallacy of affirming the consequent, which is why it shouldn't be taken as a "therefore, God exists" kind of ontological argument. However, it seems to be shifting the burden of proof from the side of the believers to the side of nonbelievers.