In reading through the books I have sitting around about vikings and Norse mythology, I came across a description of Odin as wearing a broad-brimmed hat. This, of course, I knew about, but had forgotten but given my recent studies, it has given me thought, and I don't know where to start looking for research, so I'll start here. (Somehow, I imagine it looking somewhat like Gandalf's hat. Which, in turn, tends to make me think that Gandalf looked a lot like Odin. Only with two useful eyes. And a staff instead of a spear. And no wolves or ravens.)

What might Odin's hat have looked like and was there any evidence of similar hats and headgear being worn by viking-era Norsemen? Or is it a post-Christendom invention retroactively fitted to Odin?

Anything will help! Thanks!

Well, I'm by no means an expert on Viking Age clothing. But given my current understanding, the only Viking Age head wear to be found so far has consisted of close fitting caps and some hoods. And none of them had brims.

However, several have been found with tablet woven bands at the base of the cap, and I personally think the term "brim" in the sagas may actually refer to this band of trim. If this is the case, the real image of Odin would be much less dramatic than the Gandalf image you mentioned.

i do believe in some anglo saxon manuscripts there have been depictions of saxon men wearing what appear to be rush or straw wide brimmed hats, these i most definately know existed in the roman empire and later on in the mediaeval age so the idea of it being a straw, rush or reed hat is possible.

There are a number of depictions and finds of brimed caps or hats in the medevial material. These appear to be simple felted affairs, that are not very broad brimed by modern standards.
Since this kind of hat can be made by anyone with a bit of unspun wool and some time on their hands, they have been in use pretty much any and everywhere. They have, however, never been very pretty, and thus more of a utility item than fashion.