Sorry in advance for the giant comment. But this

"Albus Dumbledore was smiling with a strange sad look in his eyes, like someone who has taken one more step toward a foreseen destination."

makes sense of this

"And Minerva made it clear to me that Hogwarts required a competent Defense Professor this year, even if I had to haul Grindelwald out of Nurmengard and prevail on old affections to persuade him to take the position."

which looked like a type 3 foreshadowing. I think Dumbledore expects and intends to die soon. Certainly we're moving in that direction, with the recent talk of McGonagall succeeding him as headmaster. It looks likely to me that Dumbledore is plotting to end his own life at the hands of Grindelwald, whose return as an antagonist to Dumbledore was foreshadowed by the story of Peter and Sirius. This, in turn, reminds me of this

Are you ready? Good. I am going to pretend to cast the Killing Curse on Professor McGonagall - DO NOT REACT, Hermione!

which looks like utter lunacy, but apparently is Dumbledore's idea of inspiration. He means to go out like Gandalf.

Probably. The alternative would be that he just thinks his time as a mysterious old wizard is almost up. Or it'll be gur Cbgvbaf Znfgre again. (Spoilers for the original novels.) But I think it's more likely than not that he'll be the one to give Grindelwald his role in the story, given that Dumbledore names himself as the one to haul him out of Nurmengard and ur ratvarrerq uvf bja qrngu va pnaba. Two birds with one stone.

In any case, in this story, Dumbledore wishes to die to escape a wasting death from advancing senility, not a phefrq unaq. To collect the relevant quotes in one place:

Professor McGonagall looked a little sad at that. "Not on purpose, Miss Granger, but I think... well, it probably is true that sometimes the Headmaster has trouble remembering what it's like to be a child."

Professor McGonagall finally spoke, and she said in a whisper, "I'm so terribly sorry, Miss Granger. I did not think the Headmaster would say such things to you. I think he truly has forgotten what it is like to be a child."

"Maybe you're right, Harry. Maybe I have forgotten over the decades what it's like to be a child."

She swallowed, hard, and said, "Mr. Potter, at thirty hours per day, you'll - get older, you'll age faster -" Like Albus.

Plus the zillion uses of the phrase 'the old wizard', and the dog that didn't bark: the fact that still nothing has been made of the connection between Dementors, Dementation, and dementia. Dementors are named for the death of the mind, both of them eat away at memories and personality, and this has gone entirely unmentioned. Even the author's notes that listed the similarities between Dementors and death left it out.

Oh, and the fact that Dumbledore's really old, and that's what happens to people when they get old.

So there's a suitably heroic reason for Hermione to research the Philosopher's Stone - either to prevent it from happening or to prevent it from ever happening again.

Edited repeatedly for clarity.