Let's start with some basic parsing. Rowling said we should have asked why Dumbledore had James' cloak "at the time James died" which is a very interesting statement. When Harry first receives the cloak, anonymously, on his first Christmas morning at Hogwarts, the note reads
Your father left this in my possession before he died. It is time it was returned to you. Use it well.
Hmmm. "Left this in my possession before he died." Had it "at the time James died." I know, they're really close, but there is just a teeny weeny bit of difference in emphasis. And it's interesting that in both cases, James' death is referenced.
So many things keep pointing us back to Godric's Hollow and the night of Harry's parents' death, the night Voldemort tried to kill baby Harry, only to have the AK curse rebound and send him into involuntary un-embodied exile (which we now know was as close as someone could come to killing him, given the dark and twisted steps V. had taken to try to ensure his own immortality).
I keep trying to piece together that night, as well as we can, given what Rowling has given us in the first six books. There are sizeable holes in our information, but it does seem clear that an awful lot of people were at (or nearby, or watching over) the Potters' house that night. Voldemort was there, of course, and Peter Pettigrew had to be closeby. We know that much because he was the Potters' secret keeper, and only by divulging where they were (and breaking the Fidelius Charm -- interesting to note that he was "breaking Fidelius" or "breaking faith") could V. have found them in the first place. James' best friend and Harry's godfather Sirius Black was their first choice for secret keeper, but apparently (?) Sirius thought he seemed the obvious choice, and convinced James and Lily to use Peter instead, a decision he regretted the rest of his life. We can also speculate that Peter had to be around in order to retrieve Voldemort's wand -- someone had to take care of it and get it back to him later, and P.P. seems the likeliest candidate.
But we know Sirius had to be somewhat nearby too, or at least close enough to rush to the scene once he heard what happened -- and how did the word get out so quickly? -- because Sirius shows up in time to loan Hagrid his flying motorbike. Sirius had come with the intention of rescuing Harry -- we know that from PoA when Hagrid is recounting the evening to Fudge and McGonagall and tells them that Sirius had said "give me Harry" but Hagrid was already there under Dumbledore's orders to take Harry away. Presumably Sirius loaned him his bike, Hagrid flew off to get baby Harry to Dumbledore (meeting him on the wall outside 4 Privet Drive in one of the first scenes of the first book). Back at Godric's Hollow, by this time the muggles have come round to see what's happening (Hagrid described the house as almost destroyed) and Peter runs into Sirius, accuses him of the crime he himself has just committed, and fakes his own death in front of witnesses (taking a number of innocent muggles with him). This leaves an enraged and unhinged Sirius behind to be taken into wizarding custody for a crime he didn't commit.
Does this sound right? I feel like I'm patching together a quilt!
And how did Hagrid get there so quickly? It's clear Dumbledore was watching Godric's Hollow. He knew the Potters' lives were in danger, perhaps better than anyone (because he knew the prophecy) so he was clearly standing guard or had someone else standing guard. Maybe the Potters' kept a portrait of Dumbledore at their house? (Sorry! Highly unlikely, but I couldn't resist.) My guess is that Dumbledore had set careful watches/guards on both the Potters' and the Longbottoms' homes since at the time that the prophecy was made, he wasn't sure which child it meant and which one Voldemort was most likely to go after. I really need to go look at the end of OotP again and revisit that whole conversation Dumbledore has with Harry about that time period. (Also I'm unclear if we know from the patched together quilt of past events exactly when Alice and Frank Longbottom were tortured into insanity. And do we know WHY they were tortured? Beyond the fact that they were on the right side, I mean? Could they have been tortured because they wouldn't reveal the Potters' whereabouts?)
Back to Hagrid. How did he get there so fast? Does he know how to appararte? His wizarding training was never completed because he got expelled, but I'm guessing he's gotten a bit of extra training on the side from Dumbledore. But if he COULD apparate, why didn't he just do so with baby Harry instead of borrowing Sirius' motorbike --unless he knew how to apparate himself but had never practiced "side by side" appartion and was too scared to do so with the baby? Could Hagrid have been on the spot at Dumbledore's orders? Dumbledore was worried; so worried about the Potters that he himself wanted to be their secret keeper. And we know how much he trusts Hagrid. I would trust Hagrid with my life. But if you were going to station Hagrid somewhere nearby, how would you hide him? Would the invisiblity cloak work for this? I know it's not big enough to cover Hagrid, but I think something similar to an "engorgement charm" might be used to make it temporarily larger.
The only problem I have with Hagrid being in the vicinity that night is that I can't imagine him just staying put and watching everything unfold without losing his temper, rushing in, bellowing like anything, and trying to take out that ruddy Voldemort all by himself. The only thing that makes sense is if he purposefully was told to stay put no matter what -- and did so out of obedience to Dumbledore's orders. We know that Voldemort can apparate, and I think it would have been relatively easy for him to do so into the Potters' house once the Fidelius Charm had broken. The Fidelius was the strongest protection -- any other magical protection on the house was likely easy to breach once the Fidelius had been breached. So perhaps Hagrid, hidden in the vicinity, didn't see anyone go in and didn't know anything was going on until he suddenly saw green, flashing light and heard screaming and shouting. And by then, of course, it was too late.
Yes, I know...Snape is another candidate for having been there that night, under Dumbledore's orders. Snape himself knew the prophecy, or at least part of it. But would Dumbledore have sent Snape, knowing how Snape felt about James? What would be the point? (Unless he also knew how he felt about Lily, but now I'm *really* reaching into the realm of speculation...) And could Snape, even though he's a brilliant actor, have seen what happened that night at Godric's Hollow (and known his own role in the unfolding of the tragedy) and remained so outwardly cold and unfeeling to Harry (for six years!) regarding his parents and his loss? If he had been there that night would that night not have become one of his worst memories, or (if you still believe Snape's bad through and through) at least a memory that he would definitely need to safeguard from Harry ever accessing? And if so, why wasn't it one of the memories he placed in the pensieve for safe keeping before he had his occlumency lessons with Harry? (Unless we assume it was there, and Harry just didn't have time to see everything.)
When I start to think about why Snape might have been there that night, one of the only things I think of is that it had something to do with trying to block the making of the final horcrux. I would bet that Snape, drenched as he was in the dark arts during his formative years, might know a thing or two about horcruxes. But did he know Voldemort was trying to make seven of them? Did Dumbledore even know that much at that point in time? I am feeling really muddied about the timeline here! Can anyone else (please!) come up with a reason why Snape, or for that matter anyone else, might have been present under the invisibility cloak?
I'm not saying that the invisibility cloak had to play a role that night, but it seems likely given JKR's words that Dumbledore's possession of it "at the time James died" is "significant" even "crucial." I've frankly wondered why Dumbledore had it and not, say, Sirius. If you had a valuable treasure like that and wanted to ensure your child would have it one day, in the event of your own likely death (considering there was a war on and the Potters were on the frontlines) who would you give it to? Seems to me you'd give it to your child's godfather. So why Dumbledore, especially since Dumbledore, of all people, wouldn't ever need it, as JKR has helpfully pointed out to us here again (and first pointed out in the chapter on the Mirror of Erised in Sorcerer's Stone, when Dumbledore tells Harry straight out that he doesn't need an invisibility cloak to be invisible)? Unless James entrusted it to Dumbledore because Dumbledore told him someone else in the Order of the Phoenix needed to use in, perhaps in part to help protect the Potters. Just how rare are these cloaks, I wonder?
A couple more notes and then I'll wrap this up. It does seem interesting when we consider other times we've seen the invisibility cloak in action. Once in PoA -- Harry carelessly leaves it at the base of the Whomping Willow, Snape puts it on and thus is able to get into the Shrieking Shack unnoticed (where he hears all the stuff Lupin and Sirius are saying about Pettigrew, etc...although I need to go back to that sometime and figure out just where Snape must have begun listening). Secondly, we see it at the end of HBP when Dumbledore instructs Harry to wear it -- then immobilizes him so that he's both invisible and not able to move during the scene where he witnesses Dumbledore's death. It seems clear here (and indeed early on in Sorcerer's Stone, when he first gives Harry the cloak) that Dumbledore is providing protection but also encouraging Harry to move forward in his daunting task of facing evil, and in some ways schooling him in some very hard and difficult lessons. I can't help but think Dumbledore, by choosing the manner of his death (at least in part) also chose to have Harry witness it for a reason. All of which leads me back to the probability that asking someone to be in a difficult and dangerous place, under the protection of the cloak, is something Dumbledore would have been prepared to do that night at Godric's Hollow if he truly felt it was necessary. Was it? And if so, who did he ask? At any rate, we know Dumbledore did ask Harry to wear it in a a difficult and dangerous place 16 years later on the night of his own death, in the same ongoing battle...and perhaps that image is a kind of full circle pattern.
So interesting this all has to do with an invisibility cloak, of all things. The end of Rowling's story seems "cloaked" for now. Although I suspect what we need to unlock it is more visible than we realize, if only we could figure out what pieces are most important and how to fit them all together!