"what kind of EVIDENCE would convince you that a God [...] indeed exists?"
See, that's the problem...
You can only look for evidence if you have a hypothesis first.
It doesn't work the other way around.
(That would be called "Texas Sharpshooter". You can look it up.)
Most of the time, people don't even know a single characteristic of the god they are proposing, which makes "forming a testable claim/hypothesis" and "looking for evidence/measuring it" nearly impossible.
Ask for a specific definition, first.
You'll see that they either have no clue what exactly they are trying to argue, or that they want you to come up with something, which is simply not your duty, but theirs (the burden of proof lies on them, not on you).
"[...] If you were a God, how would you try to prove your own existence to your rational beings?"
A god, by my definition would be able to pull that one off quite easily.
He could make it so that we all believe in him.
Otherwise he would be pretty incompetent, seeing that he probably caused us to exist in the first place (which would depend on the definition of "god").
"how would you go about convincing such a person?"
Tell him that -your- god talks to you, telling you that his god is a false one.
One that tries to occupy his mind.
You can have all kind of good fun with this one.
He will either accuse you of lying, which you can counter pretty easily, by simply saying the same (you need to stand your ground and always keep up a straight face for bonus points), or he will end up being mindfucked, seeing that there might in fact be some other being talking to him (which is probably less likely to happen. People can be pretty emotionally invested in their hallucinations).
Either way, you can pretty much mess a little with him.
If he really hears voices, he suffers from schizophrenia.
Some therapy would be appropiate.
And no, schizophrenia and therapy is nothing that should cause some kind of social stigma.
It's just a psyche that doesn't function the way it should function.
Yet people feel somewhat embarrassed or offended, when you address psychological issues for some reason.
It's more like the flu or a broken leg to me. It's something that can be fixed for the most part and shouldn't prompt defensiveness or the breaking of one's fragile ego.
If he wants to keep his voices, that's fine.
If he ever takes them as a justification for immoral or harmful things, call him on his bullshit or call the police, if things get out of hand.
People need to take responsibility for their own deeds.
In most cases, though, people simply claim to hear the voice of god or something like that, because people give them attention, if they make such outrageous claims.
If god really speaks to him, ask him some questions, like the lottery numbers that are going to win next.
A god that can create a whole universe with every star, supernova and planet in it should know such a rather trivial thing, shouldn't he?
You can use a ton of other questions if you want some combo-points.
In any case: I would call him on his bullshit, once you proven with those tests that he's talking nonsense and that he's probably doing it for the attention.
Let him know that you won't fall for this silly crap and the he should act like an adult, for goodness sake.
