Religion might rebut atheists by describing a concrete phenomenon accessible to anyone, that is consistent with religion but cannot be accounted for by science. Candidate: the ability of faith to manipulate events.
Example: I have a problem, that is, something about the world I would prefer to be different. To solve the problem I can act in the world, moving the furniture around in order to actualize my solution. Or I can resist the urge to change things and instead leave it to God to decide how to solve my problem.
Millions of religious for thousands of years, myself included, have claimed that God can and will solve our problems better than we can (and without all the effort involved in furniture-moving), but only if we get out of God's way and let God decide how to solve the problem. (Or even whether to "solve" it at all: I may have mistakenly labeled something a problem when I am in fact better off with it than without it.)
I've tried it. It works. Physical science cannot explain how an emotional state entirely within myself can manipulate physical objects and events outside myself. But I've seen it work so many times that now, when faced with a problem, I resist the urge to do something about it, and instead accept it as God's will unless and until God wishes to change it. The solutions God then delivers are so elegant, not to mention effortless, that I try now to consign even the largest decisions in my life to God. I literally bet my life on it.
At the same time I have seen how often my activist efforts to change the world by doing something to solve my 'problem' are frustrated and thwarted by unforeseen circumstances, as if the effort itself were counterproductive and self-defeating, succeeding more in undermining my solution than in actualizing it.
To me this explains why violence (eg, war) has never worked as well as one always expects it to: violence should be the MOST effective way to force the world to conform to my wishes because it is the most determined and most forceful. And yet violence never seems to accomplish what it intends, as often as not aggravating the problem instead of solving it.
Similarly, if faith in God did not work, it would be the worst problem-solving strategy because the faithful would inevitably be overwhelmed by all the problems they faced but did nothing about. Faithful deference to God ("Thy will be done" or "Insha'allah") would be evolutionary suicide. And yet the faithful thrive.
See: Jacob's Mirror: A Reconciliation of Science and Religion at YouTube or Amazon.com. #scienceandreligion