Listening to a recent exchange between Richard Dawkins and A. C. Grayling, and reading Grayling's subsequent email to-and-fro with Jerry Coyne, I've realised that there's probably nothing that would convince me of the existence of God.
This of course stems from my naturalistic worldview, in which supernatural entities or events are precluded by definition. However:
There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy.
William Shakespeare: Hamlet, Act 1, Scene 5
The Bard may have been suggesting that there are things we know nothing of, but the more we find out, the more we realise that the universe is strange. And though strange, it has never, not even once, been found to be supernatural. Every time a "mystery" has been explained by science, that explanation has been a natural one. Not once has science come up with an explanation involving supernatural forces. Some might object that the remit of science prevents this, but what else are we to go on? Divine revelation?
Before I'm accused of being closed-minded on this issue, I should first point out that I consider the problem to be essentially one of definition. If you ask me what it would take for me to be convinced of the existence of God, I will necessarily want a clear definition of that God. You tell me precisely and coherently what you mean by God — with no obfuscation or appeals to mystery or ineffability — and I will tell you precisely and coherently what it would take for me to believe in that God.