How do you convince a non-believer that God is real? Fight them on their terms. Atheists like rationality, experiments, data and hypothesis testing. So devise a study that has predictive validity. It's that simple.
Make a specific verifiable prediction about the future whose outcome is not subject to vague or multiple interpretations apart from a religious one. A specific prediction that is so unlikely to occur except in the event of supernatural intervention. And if it were to occur then the only possible inference would be the supernatural. This is your hypothesis. Then devise an experiment to test this hypothesis, with controls.
For example, here's one hypothesis to prove that God is real and he answers prayers - over time t, a clear majority of n number of people suffering from a terminal illness who pray to be healed, will be healed from that illness. As a control, we can compare the results from this study to those from a group of n number of people with a terminal illness who do not pray to be healed. If the death rate is significantly lower in group A, then prayer works and God is real, if it is more or less the same in both groups then prayer has no effect and God is less likely to be real.
Also, the number of people who pray to be healed and die, or the number that don't pray to be healed and live should be so minuscule as to be negligible. In other words, for us to know that prayer works, the majority in both praying and non-praying groups should be true positives and true negatives respectively. And the number of false positives and false negatives should be tiny.
This is a good hypothesis because it separates a real effect from what would be chance or randomness but which might be confused for something else by believers.
One person being healed after prayer is not valid or reliable evidence in itself. It could be due to a number of factors, some of which could be as yet unexplained. A true effect would be visible over a group of people over time under similar conditions. And even if this happened, if a large number of people got better repeatedly after praying, you would still need to compare that effect to a similarly large group of people who got better repeatedly over the same time period without praying. By controlling for this one variable, you would ensure that prayer alone and not other confounding factors were at work.
No rationalist will every accept the results of a study as proof of course. You never 'prove' anything in science. You merely suggest that something is more or less likely. You goal here is to conduct an experiment that demonstrates that obtaining the results you obtained by chance alone would be so unlikely as to be as good as accepting that God exists (or perhaps an alien or the Matrix posing as God). This is as close that any scientist will come to believing in God.
I have yet to come across any data that shows that prayer works. All the evidence I am usually shown is no different from statistical noise (false positives or negatives) rather than true effects. But the experimental design is robust and I'm willing to change my beliefs and believe in God if that's what the model reflects. But we all know that's never going to happen.
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Counter claims like God shouldn't be tested, God chooses not to be tested, God does what he will, you can't understand the mind of God, God has a larger plan, etc. to explain why prayer doesn't always work are all invalid and pointless as they don't support God's existence. They merely offer data that is no different from random chance or statistical noise (if not worse), making you question why you would choose to believe in God to begin with. If all the evidence for God is so inconsistent as to be no different from evidence for no God, then why hold to such a theory?
And the argument that you're trying to measure something that isn't measurable or isn't supposed to be measurable is of course rubbish. That's the point of doing the measuring. If you can't measure an effect, then you have no reason to believe in it. And if you still believe in it, on what basis?
Of course, a true believer doesn't trouble himself or herself with such trivial concepts like evidence. The best way to Know is to surrender yourself to the unknowable, to have the humility to know that you will never understand what you were never meant to understand, to take a leap into darkness, which is a good way to delude yourself into believing anything, true or false, and which also comprises logical fallacies like circular reasoning and false premise reasoning by first assuming something is true in order to believe in it.
While this may bring comfort to people it is still the equivalent of a logical fallacy, and if something is the equivalent of a logical fallacy, it is as likely to be true as untrue. Which is why I default to not believing and people who prefer comfort default to believing. Two different frameworks for making sense of the world. Ironically, it's the scientific one that actually does more to prove the existence of the supernatural. Which is why we build verifiable models to make sense of the world. Religious models are just lazy. They explain too much by being deliberately vague. Great for self-delusional comfort, not a good way to derive insight.
