When it comes to the question of what would change your mind on a particular issue, it's meant to be a sign of an open mind to be able to come up with some form of evidence that would persuade you. But that standard is problematic, especially in cases when there's philosophical problems. For example, what evidence could convince me that the world started last Thursday in its current state as if time had passed? I'm not sure there's any evidence that could persuade me of that or the contrary. There's just no evidence that could satisfy that conclusion.
When it comes to the question of God I used to be all about the evidence, but I've been swayed away thanks to Massimo Pigliucci and PZ Myers, and now I don't think evidence could be sufficient. I think I need to make a distinction, however, because it's not saying my mind is made up and nothing can persuade me.
Firstly, I think there can be evidence of something, if there were interventionist agents affecting the world then that should be visible. That there's not good evidence of this counts against the notion of God. I suppose one could appeal to being able to work in undetectable ways, but that means that God is indistinguishable from no God at all.
However, this does not mean that evidence can be sufficient to establish God in the supernatural sense because the supernatural is not a coherent notion. Furthermore, we could not establish that God is 'uncaused', 'simple', 'eternal', 'immaterial', 'omnipotent', 'omniscient', etc. as all of those are beyond what is even possible to establish by evidence. Thus even if we did have evidence of something, it would be a leap of faith to say that it's God.
It's not a question of ontology but one of epistemology. Assuming God exists in the way that is described by theologians, we can never satisfy those attributes through scientific investigation. I think something god-like is hypothetically observable, but to say it's God is making unwarranted inferences.