If you want to evaluate their claims fairly, you should try to understand what arguments and circumstances were most influential to them.
If someone doesn't pursue this question about themselves, for themselves, it's useless for someone else to (merely) ask it, as even if somehow you learn the truth, it probably won't have the form of a good argument. Arguments need to be designed, they won't describe the structure of a system as complicated as human mind (with influences of culture), unless there are laws of reasoning firmly in place that ensure that beliefs (by construction) result from arguments.
When a person becomes skilled at careful (correct/useful) argument, that point is reached already in motion, with some of their beliefs formed by something other than careful argument. After epistemic habits become moderately healthy at working on new questions, they aren't automatically healthy enough to purge the mind of convictions that were previously put in place by less reliable processes (and studying the details of those processes isn't necessarily a good use of your time; just ask the questions again, not forgetting the question of whether the questions are important enough to work on, to prepare beliefs about).
(In particular, God-related issues seem to be primarily a problem of distorted relevance. If you only observe the world and try to understand it, any specific supernatural hypothesis (i.e. with well-designed meaning) won't become important enough to worry about. So a healthy way of discarding God, if you find yourself affected, seems to be about realizing that the meaning of the concept is unclear and there is no particular reason to focus on any clarification of it, rather than about falsity of related arguments.)
rather than about falsity of related arguments
E.g., you could check whether there's anything of pragmatic relevance in ideas like the convertibility of transcendentals. Just ditching convertibility of transcendentals can easily imply a God that is not similar to the God of e.g. Aquinas, to a lesser extent Aristotle, to a lesser extent Leibniz. This doesn't take much time and quickly rules out large portions of the "God" conceptspace. (Also, people who believe in vague "God"s (e.g. "God isn't a person, God is the Goodness inhere...
I am about to graduate from one of the only universities in the world that has a high concentration of high-caliber analytic philosophers who are theists. (Specifically, the University of Notre Dame, IN) So as not to miss this once-in-a-lifetime opportunity, I have sent out emails asking many of them if they would like to meet and discuss their theism with me. Several of them have responded already in the affirmative; fingers crossed for the rest. I'm really looking forward to this because these people are really smart, and have spent a lot of time thinking about this, so I expect them to have interesting and insightful things to say.
Do you have suggestions for questions I could ask them? My main question will of course be "Why do you believe in God?" and variants thereof, but it would be nice if I could say e.g. "How do you avoid the problem of X which is a major argument against theism?"
Questions I've already thought of:
1-Why do you believe in God?
2-What are the main arguments in favor of theism, in your opinion?
3-What about the problem of evil? What about objective morality: how do you make sense of it, and if you don't, then how do you justify God?
4-What about divine hiddenness? Why doesn't God make himself more easily known to us? For example, he could regularly send angels to deliver philosophical proofs on stone tablets to doubters.
5-How do you explain God's necessary existence? What about the "problem of many Gods," i.e. why can't people say the same thing about a slightly different version of God?
6-In what sense is God the fundamental entity, the uncaused cause, etc.? How do you square this with God's seeming complexity? (he is intelligent, after all) If minds are in fact simple, then how is that supposed to work?
I welcome more articulate reformulations of the above, as well as completely new ideas.