Can a coincidence be evidence of a god?

It depends on what you mean by evidence. According to evidential epistemology, anything that can be construed as evidence can be used to justify a conclusion. So, are coincidences in and of themselves not a form of evidence? I know of no rule that rules coincidences, arbitrariness, and randomness out as forms of evidence. And the key is 'can'. I would say it is material fact that people regularly choose to make coincidence into evidence to support their beliefs. Consider a famous cognitive bias called illusory correlation. In it, what in actuality is coincidence is viewed as correlation. This is so common, there's an expression in English for it: "giving a sign" (mindbodygreen.com). Throughout Christian history, people have been "given signs from God".

So, really, since coincidence is regularly taken to be correlational evidence, it is not a matter of 'can'. It can and does occur regularly. Your question should be written:

Should an apparent correlation taken as evidence be reduced to coincidence to reject it as evidence of a god?

And the answer depends on who you ask. A contemporary skeptic (IEP) will argue it is VERY important, given the bias of illusory correlation to reflect heavily on what probability means, and to consider that correlation is not causation, and coincidence is not correlation. Our brains routinely fool us, and so it's best to have very thorough epistemological knowledge and method to sort out coincidence, correlation, AND causation. This sort of thinking is typical of fallibilism (IEP) and naturalistic versions of scientific practice where everything, with prima facie cause, can and should be doubted. That's exactly why science takes an observation, and asks questions like "is this a coincidence or correlation" and then sets about trying to devise tests to prove things empirically one way or another. Mere a priori introspection and rational methods are simply too likely to mislead a thinker.

Can a coincidence be evidence of a god?

It depends on what you mean by evidence. According to evidential epistemology, anything that can be construed as evidence can be used to justify a conclusion. So, are coincidences in and of themselves not a form of evidence? I know of no rule that rules coincidences, arbitrariness, and randomness out as forms of evidence. And the key is 'can'. I would say it is material fact that people regularly choose to make coincidence into evidence to support their beliefs. Consider a famous cognitive bias called illusory correlation. In it, what in actuality is coincidence is viewed as correlation. This is so common, there's an expression in English for it: "giving a sign" (mindbodygreen.com). Throughout Christian history, people have been "given signs from God".

So, really, since coincidence is regularly taken to be correlational evidence, it is not a matter of 'can'. It can and does occur regularly. Your question should be written:

Should an apparent correlation taken as evidence be reduced to coincidence to reject it as evidence of a god?

And the answer depends on who you ask. A contemporary skeptic (IEP) will argue it is VERY important, given the bias of illusory correlation to reflect heavily on what probability means, and to consider that correlation is not causation, and coincidence is not correlation. Our brains routinely fool us, and so it's best to have very thorough epistemological knowledge and method to sort out coincidence, correlation, AND causation preferably in the formalisms of statistics and probability. This sort of thinking is typical of fallibilism (IEP) and naturalistic versions of scientific practice where everything, with prima facie cause, can and should be doubted. That's exactly why science takes an observation, and asks questions like "is this a coincidence or correlation" and then sets about trying to devise tests to prove things empirically one way or another. Mere a priori introspection and rational methods are simply too likely to mislead a thinker.