Here is a commonly cited example:
I was healed from cancer by God!
Really? Does that mean that God will heal all others with cancer?
Well... God works in mysterious ways.
A key characteristic of ad hoc rationalizations is that the "explanation" offered is only expected to apply to the one instance in question. For whatever reason, it is not applied any other time or place and is not offered as a general principle. Note in the above that God's "miraculous powers of healing" are not applied to all cancer sufferers, but only this one at this time and for reasons which are completely unknown.
In the above, the idea that not everyone will be healed by God contradicts the common belief that God loves everyone equally.
How could we tell when it is happening and when it is not? How could we differentiate between a system where God has acted in a "mysterious way" and one where the results are due to chance or some other cause?
i) I disagree with the setup. Many atheists, as well as some Christians, routinely recast all truth-claims in terms of evidence and counterevidence. No doubt that's appropriate in cases where there is both prima facie evidence and prima facie counterevidence, but everything shouldn't be hoisted onto that that seesaw.
ii) For instance, we often believe sometime happened based on direct evidence that it happened. I believe certain things happened to me because that's a matter of personal experience. I don't put that on one side of the scales, put possible counterevidence on the other side of the scales, then see which way the scales tip. That's very artificial. I simply believe it happened because it happened to me, and, in the nature of the case, I have firsthand knowledge of things that happen to me.
Likewise, we believe lots of things based on what trusted people tell us. We don't ordinarily feel the need to counterbalance that belief by considering possible evidence to the contrary, then decide if one outweighs the other. The teeter-toter paradigm doesn't fit our general belief-forming system, or even the justification of beliefs.
iii) Why does God not healing somebody else equally deserving furnish any kind of evidence that God didn't heal me? What's the connection? If there's evidence of divine healing, why isn't the evidence in itself the only salient consideration?
Suppose, unbeknownst to me, cyberterrorists hack into the traffic light system to facilitate a bank heist. On the one hand it gives the getaway car an escape route. On the other hand, it blocks traffic on the same side of the street where the police station is located.
However, that has the fringe benefit drivers in my lane have solid green lights all the way home, while drivers in the opposing lane, and side streets, have solid red lights. In my ignorance, I have no idea how to account for the disparity. Moreover, this is something extraordinary.
Yet that doesn't count against the indisputable fact that, for some inexplicable reason, the traffic lights favor everyone in my lane. They just do! It may cause me to investigate why that's the case. But it's not the phenomenon itself that's in question. That's not a reason to doubt that on this particular day, the traffic lights in my lane stayed green all the way home. And that's not a reason to doubt that it requires a special explanation.
iv) In addition, the objection presumes, without benefit of argument, if God heals people at all, we'd expect him to heal all equally deserving people. But is that a reasonable expectation? What's that based on? Just that it seems arbitrary for God to heal some, but not all, equally deserving people?
But it's not hard to come up with reasons why that might be so. Consider the alternative: suppose God healed everyone who prayed for healing, or everyone who was prayed for. Well, that would change the future, in the sense that the future would turn out very differently in that event than if God didn't heal everyone. Who lives and who dies, where they live and die, when they live and die, affects the future. If more people live longer, that has multiple ramifications.
So one reason God might not answer every prayer for healing is because that's inconsistent with the future he intends. For instance, some people die because other people didn't die. Take a terminal cancer patient who's miraculously healed. A year later, he kills a cyclist or pedestrian while driving under the influence.
It sounds swell to say God should heal everyone, but what is good for one person may be bad for another. Your healing may come at someone else's expense, down the line. Something you do today may unintentionally harm someone tomorrow.
On the other hand, one reason God might heal some people is to furnish evidence for his existence. He performs miracles often enough to maintain a periodic witness to his existence, but he refrains from performing miracles routinely because that would result in a very different future.
v) Incidentally, I, as a Calvinist, reject the premise that God loves everyone.