At its core, this topic boils down what is meant by "evidence."
I define evidence as "Anything that lets us reliably differentiate between ideas that are merely imaginary, and ideas that accurately describe reality."
So, if you propose some method, framework, or argument as evidence, but that same framework can equally support a contradictory conclusion by replacing some nouns in your argument, then this wouldn't count as evidence. For example, you can propose an eternal necessary god as the ultimate foundation of reality, but I can propose with no more or less evidence an eternal necessary aspect of nature instead, with brute-fact mindless properties that cause it to create the world exactly as we see it.
The best framework for creating reliable evidence that humans have identified is the scientific method. In particular, novel testable predictions that are tested and confirmed. If you think a different method also works, just demonstrate that and it will become part of science.
The scientific method outputs evidence that can support one predictive model while excluding that model's explanatory competitors. A "model" is a conceptual framework about how some aspect of reality works. Evidence is provided when a new prediction made by a model, that no one has tested before, is subsequently tested and confirmed to be accurate. A supported model is more likely to be objectively true than the unsupported models, but note that there is no "proof" or "truth" or metaphysical certainty. It simply has demonstrated that it reflects a deeper understanding of some part of reality.
In science, a very very supported model, which has made many new and accurate predictions, becomes a "theory." While every model is imperfect and has explanatory boundaries, the fact that they can predict things we don't know yet about reality is the hallmark of science and the method guiding its discoveries about the facts of reality. This is why we have airplanes, computers, and medicine today.
So far, theism as a model, as a conceptual framework about reality, has no evidence to support its accurate correspondence to reality. It also has no supporting evidence produced by the scientific method, and so it can't be meaningfully distinguished from the infinite array of other false imaginary models that don't accurately describe or correspond to reality. It could still be true, just like any of the other imaginary models have a tiny chance to be true, but that is a useless possibility because it is functionally zero. Theism in general isn't even a predictive model, just a post hoc sufficient explanation of the things we already knew; its various forms don't typically rise to the level of a testable hypothesis.
From everything I've ever seen, forms of evidence that theists propose are unreliable and/or fallacious. A fallacy is a way of reasoning, identified by logicians, that reliably leads to false conclusions. Another way to say this is that the arguments are not valid in logical structure, or the premises are not sound because they cannot be demonstrated. Here are some common argument forms proposed as evidence by theists, which are in fact not evidence:
1. "I believe that the book is true because the book says it contains perfect truth, and says it cannot be wrong." This is circular reasoning, a fallacy. It is not evidence because any other contradictory book making similar claims is equally supported.
2. "I define morality as coming from a god. We have morality. Therefore god exists," or "I define god as existing necessarily, therefore god exists." Definitions are not evidence. They describe our thoughts about abstract ideas in our heads. This is imagination. Nothing tethers it to the actual facts of reality.
3. "We don't have an explanation for X. I can imagine a thing G that, if it existed, would sufficiently explain X. Therefore this is evidence for G." This is not evidence, but a fallacy called the Argument from Ignorance. For any phenomenon, there are an infinite number of explanations we could propose that would sufficiently explain the phenomenon. The proposal by itself, of any one of these infinite explanations, is not evidence.
4. "I just can't imagine how life could arise without a guiding intelligence. A supernatural creator feels like the most probable explanation." This is an Argument from Incredulity, a fallacy. Your failure of imagination is not evidence, nor is your imagination evidence for reality. If you have an idea for an explanation, you need to show that it corresponds to reality with specific supporting evidence.
5. "I intuitively feel that god exists," or "I don't know how I would get out bed each morning if I didn't believe in god," or "I'm afraid of dying and god gives me hope for eternal life." Emotional appeal is not evidence that something is true. Subjective feelings are not evidence. Different people have wildly different and often contradictory emotional needs, intuitions, and hopes, depending on their culture and personality.
6. "Look at how influential religion X has been throughout history," or "look at how many people believe religion X." This is called a bandwagon fallacy. Different popular beliefs hold sway at different times, many of which are contradictory or have been proven factually wrong.
7. "I had an emotional experience at summer camp that one year, that I can't explain, and I know it was god filling me with love." Again, this is an argument from ignorance. Religious indoctrination primes people to interpret emotional feelings a certain way, but this is not evidence because nothing specifically supports this explanation over any other.
8. "We prayed and someone's cancer disappeared" or "I converted and my drug addiction went away." Anecdotes are not evidence. The plural of anecdote is not data. There is a known rate of spontaneous remission for cancer, which occurs regardless of praying or not praying, or the religion of the patient. It occurs at similar rates in animals that have cancer. Likewise, people overcome drug addition through hobbies, marriage, divorce, and many other routes. There is no data that shows religiosity affects these ailments at a higher rather than anything else.
9. "I found a published paper that says light cures cancer, or consciousness is immaterial, or the earth is flat." Every group of experts includes a fringe minority with ideas that are not accepted by the majority consensus of experts in their field. A fringe view is not evidence. Publication of a finding does not make it evidence by itself; rather, the ability of the published finding to persuade the majority consensus of experts in the field is what transforms this finding into scientific evidence. This distinction, and this appeal to the scientific consensus, is often falsely labeled as a fallacious appeal to authority. However, this is only a fallacy when the proposed authority is not an actual authority (e.g. 90% of dentists agree that the Vikings are the best football team).
10. "I just have faith." Any idea can be believed on faith, therefore faith is not a reliable tool to distinguish false ideas from true ideas. Faith in the face of good evidence to the contrary is irrational.
Ok, that's enough for now! Sorry for the long post, but I hope this can be helpful for people. There are a lot of common arguments I didn't include, but they are also fallacious, unreliable methods for distinguishing imagination from reality.
