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Atelos comments on Harry Potter and the Methods of Rationality discussion thread, part 20, chapter 90 - Less Wrong Discussion
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It seems to me that Harry was a bit too quick to dismiss the Resurrection Stone option. Certainly if it functions according to his current conceptions of it it won't bring Hermione back in the sense he finds meaningful. However the experience of that soul/magic explosion at Hermione's death gives at least some evidence of a soul actually existing, even if still not enough to make it the most probable explanation for the stone's function, and there are other non-soul requiring ways that the stone could function such as looking back in time for the most recent functioning mind. Given the potential difficulty in finding it and the legend about how it's actually counterproductive its still probably not worth spending much effort pursuing it if you don't already know that pursuing it = convincing Riddle/Quirrelmort to go fetch it out from whatever defences he has it under or breaking them yourself, but he should still probably have put a bit more thought into it before rejecting it.
The impression I got from canon is that it works exactly as HJPEV believes even in the Rowlingverse; there is some evidence against this (the resurrected marauders insisting that death doesn't hurt, LIMBO! Dumbledore's information might not be things Harry could have figured out on his own), but as I recall, one of the resurrected ones said "We're part of you, after all", and Dumbledore's "Well of course it's in your head! But why should that make it any less real?" Even if there is an afterlife in the Rowlingverse, it seems like she really did not intend for there to be any method of communicating with it.
Of course, HJPEV does not have access to a copy of Harry Potter and the Deathly Hallows, so you're right that he's privileging his hypothesis, and should at least do the obvious test--come up with information that Hermione would know that Harry couldn't figure out on his own, use the stone and ask the Hermione that appears about it.
Made trickier by the fact that the information needs to be reliably checkable by Harry. And preferably not something he might already have known but mostly-forgotten.
He needs, in other words, a problem in class NP but not P. (Where P is "Potter" rather than "polynomial".)
"The location and description of a specific item in your muggle bedroom" or "Chapter, author, title and quotation from a book that you read but I did not" and "details on your family that you haven't yet told me about" seem like places to start. (The bedroom/family ones require he didn't get any such information over the Christmas break, but from the sounds of it he didn't.)