Case studies are similar in my mind to anecdotes or eyewitness testimony, an interesting starting point or indication that something might be worth digging into but not really evidence.
And yeah I suggested a double blind study because it has the most value for providing potential evidence although even that is no guarantee depending on the experiment design. It'd definitely be a good start though at the very least. You could do a non blind study but then the fact that it's non blind will be the first thing to come up and cast doubt on the results. If you want to provide solid evidence I don't think you would want to settle for less than that if you can avoid it.
FWIW I think there have already been studies done on prayer but they don't seem to be conclusive from what I could tell at least but hey, I'm not a scientist. You just asked what I'd consider evidence so hopefully this has helped answer that somewhat. Even with a double blind study though I think you would have some work ahead of you but you'd definitely have my interest!