"Why did Dumbledore have James' invisibility cloak at the time of James' death, given that Dumbledore could make himself invisible without a cloak?"
I've read theories, I've read every bit ever said about invisibility cloaks, and I've thought until I'm all thought out. I've learned a few new things--like, did you know that invisibility cloaks are woven from the hair of an ape like creature called a "demiguise"? That's why the cloaks are so rare, you see. When a demiguise realizes it's about to be caught it becomes invisible. (See "Fantastic Beasts And Where To Find Them")
After all that thinking, I've come to the conclusion that, much like the "Why didn't Voldemort die?" question that was all the buzz just prior to Book 6 (answer, "He had horcruxes"), we're missing some information that is needed to properly determine why Dumbledore had James' invisibility cloak. Nonetheless, I have two very vague, and not very creative ideas.
- The most obvious thing would be for someone in the Order to have been using the cloak, much as the current order uses Moody's cloak. This doesn't seem so extraordinary as to be "crucial" to the plot on its own. However, just maybe, the person using the cloak was somewhere doing something that will ultimately ensure Harry's success, thereby making it crucial.
- When asked, "Was there anyone present in Godric's Hollow the night Harry's parents were killed?" Jo all but answered yes when she said, "No comment." So, I think that someone, possibly Snape, was under the cloak at the Potter's.
Not very good, I know, but I'm afraid it's all I've got.