Note: First is my original theory and then after "Update" is my revised theory.
JK Rowling recently updated her website to include the following entry:
... which means, 'never asked question'.
Why did Dumbledore have James' invisibility cloak at the time of James' death, given that Dumbledore could make himself invisible without a cloak?
Prior to posting this I had a quick look on-line, and realised that some fans have been speculating about this question. However, nobody has ever asked me about it, and they really should have done. Just to allay the fears of the justifiably suspicious, this isn't what we in the know call 'a Mark Evans situation.'* There IS a significant - even crucial - answer."
So what is the crucial expanation?
When Harry received the Invisibility Cloak for Christmas in PS/SS, the attached note read:
Editing to add an explanation for how Figg could have known about the location:
Flitwick said the Secret Keeper must “choose to divulge” the location, but that doesn't mean the Secret Keeper must give the location in person or aloud. Dumbledore was the Secret Keeper for the Order of the Phoenix's headquarters at Number Twelve Grimmauld Place. Harry didn't learn this directly from Dumbledore; he learned it from a note Dumbledore had written and entrusted with Alastor Mood. So a similar situation explains how people like Dumbledore and Hagrid knew where the Potters were staying even though they didn’t know Wormtail was the Secret Keeper.
And since Sirius had engineered the switch and wanted it to work as a bluff, Sirius was probably showing people like Dumbledore and Hagrid the note indicating the location of the Potters’ house. So long as Wormtail wanted someone to know where the Potters were staying (or gave permission to tell them), Sirius would have been able to show them the note just as Moody showed Harry the note saying the Headquarters for the OotP is 12GP. So there is no difficultly understanding how Figg could have known where the Potters were staying. If Dumbledore asked Sirius to let her know, Sirius would only have needed to get permission from Wormtail. And in order to maintain the bluff, Wormtail wouldn't have refused.
In looking carefully at the 24 hours and passages in PA, I noticed Sirius said he had "arranged" to check on Wormtail that night at his hiding place (“I’d arranged to check on Peter, make sure he was still safe"), so Wormtail was expecting Sirius on the night of October 31and that seems to be THE reason why the attack was planned for that night. If Sirius was expected at Wormtail's hiding place around 11:00 PM, then Voldemort and Wormtail planned to be at Godric's Hollow at 11:00 PM. That way Voldemort would know Sirius wasn't with the Potters when he attacked and Sirius would be expected to Apparate to Godric's Hollow as soon as he realized Wormtail wasn't in his hiding place, and as soon as he arrived, he would be killed by Wormtail who was waiting for him outside. That way, the WW would wake up to find the Potters and their presumed Secret Keeper dead by Voldemort's hand, a situation that would allow Wormtail to maintain his cover as a spy in the Order. However, the curse backfired on Voldemort and Sirius flew the motorcycle instead of Apparating, so all Wormtail could do was grab the wand and figure out a way to fake his own death and frame Sirius for it. As a commentor noted, Wormtail could have been hiding in rat form, so he didn't need an Invisibility Cloak.
The reason this blows my original Figg theory is that Wormtail would, of course, know that Arabella had been given the Potters' location, and he would have killed her first so she wouldn't be able to interfere once Voldemort and Wormtail arrived at the Potters' house. I don't believe Voldemort would have wanted any additonal people to know the Potters' location, and it's for that reason I don't believe he involved Snape in his plans that evening (the hot theory is that Snape was there under James's Invisibility Cloak). The fewer people who knew the plan, the better, just as Voldemort hadn't enlisted the help of any DE's in PS/SS when he possessed Quirrell and just as he kept the number of DE's "in the know" in GF to the barest minimum (Wormtail and Barty Crouch, Jr.). I can think of no good reason why Voldemort would want Snape to know about the plan to attack the Potters and be given the Potters' location via Wormtail (by a method by which Snape didn't know Wormtail was the Secret Keeper), or if he did, how it was that Snape was at Godric's Hollow that night at the time of the attack with an Invisibility Cloak, let alone James Potter's Invisibility Cloak. Travis has posited a theory over on Sword of Gryffindor, but while it is techically plausible, I don't find it remotely likely. If Snape had been told the Potters' location and told of the plan to attack them that night, Snape could have sent a Patronus to Dumbledore, who could have alerted the Potters by Fawkes or his own Patronus. There just seems no remotely likely way to explain how Snape happened to have James's Invisibility Cloak that night or why Snape would have been trying to warn James instead of letting Dumbledore do that (for one thing, Snape would have to explain to James how he knew Voldemort was about to attack, and finding out Snape was a Death Eater would hardly have given James reason to trust a man who was already his enemy).
It is still possible that Arabella was monitoring the general area, heard/saw the destruction of the house, went in far enough to see James's body, and then sent an owl to Dumbledore saying she thought the Potters were all dead. [An alternate explanation for how Dumbledore knew is that Lily, running for Harry, sent a Patronus messenger to Dumbledore saying Voldemort had found them.] Dumbledore didn't know the Potters' location, but he would have been able to send Fawkes to the site instantly since Hedwig was able to deliver letters to Ron, Hermione, and Sirius when they were all at 12 Grimmauld Place under the Fidelius Charm. Fawkes is able to communicate enough for Dumbledore to have ascertained that Harry had survived. Hagrid could have been dispatched to Godric's Hollow by portkey with a broom to fly Harry to Little Whinging and not seen Arabella since she would have returned to her house to await instructions. So it is possible that Arabella was generally monitoring the neighborhood using James's Invisibility Cloak and had informed Dumbledore of the attack and the destruction of the house without realizing the baby had survived.
The Figg-as-neighborhood-monitor isn't a dramatic theory, but it would explain why Dumbledore had been given James's Invisibility Cloak before being killed and it's a significant--even crucial--explanation because it was the way Dumbledore found out about the attack on the Potters so quickly and was able to get Hagrid there before anyone else. Nevertheless, it still isn't a satisfying theory because it seems to me that if Figg had been able to tell Dumbledore that the house had been destroyed, Dumbledore should have known that the Fidelius Charm was broken. Hagrid hadn't even known that Sirius was the Potters' Secret Keeper, and Dumbledore sent him to get Harry out of the ruins, so Hagrid was able to find the house even without having been told the location before the attack on the Potters. So why would Dumbledore have sent Fawkes instead of Apparating out of the castle with Fawkes as he had done in OP when Umbridge and Fudge were about to arrest him over Dumbledore's Army? And although Dumbledore trusted Hagrid with his life, Hagrid couldn't Apparate legally even if he had been given private lessons (he had been expelled from school at age 13, so he wasn't supposed to do any magic), and surely saving this baby was uppermost in Dumbledore's mind. So why didn't Dumbledore take five minutes to get Harry and return to Hogwarts so he could complete his preparations for the spell on the Dursleys and do whatever else he needed to do with the baby safe at Hogwarts? Why not send McGonagall to Apparate to get the baby? Dumbledore wasn't expecting Hagrid to show up at the Dursleys until nearly midnight on November 1, so he must have believed Hagrid was taking very slow transportation (like a broom) from quite a distance.
We don’t know what time Hagrid spoke to McGonagall or where Dumbledore was when they spoke. Maybe Dumbledore was on his way out when he told Hagrid to go to Godric’s Hollow to get Harry or maybe he told Hagrid he didn’t want to be disturbed and would meet Hagrid and Harry later at 4 Privet Drive, Little Whinging. Perhaps Dumbledore was checking information related to the blood magic he was about to invoke on the Dursleys and/or writing the letter to them about what had happened and/or notifying the Order, etc. Perhaps he went to Privet Drive to set up the charm, which could be an “immensely complex spell” just like the Fidelius Charm. As Travis noted, by the time Dumbledore did appear at the Dursleys, the charm had been set up; he only needed Petunia to take in Harry to seal it.