What is the significance of the hat in Miller's crossing?

The hat is a tangible representation of Byrne's dilemma. The point to be noted is that his dilemma has so much prominence that it deserves a character of its own. There are many scenes in the movie when the hat disappears at moments of clarity and magically reappears on the stairs, exactly at the birth of some new confusion.
Choosing a hat was also a very wise decision as it is something very personal to a character, to the extent that it could define a person, just like the confusion in his head.

Just my tuppence worth

A scene in the movie provides a lovely explanation of the hat:

Verna: What're you chewin' over?

Tom Reagan: Dream I had once. I was walkin' in the woods, I don't know why. Wind came up and blew me hat off.

Verna: And you chased it, right? You ran and ran, finally caught up to it and you picked it up. But it wasn't a hat anymore and it changed into something else, something wonderful.

Tom Reagan: Nah, it stayed a hat and no, I didn't chase it. Nothing more foolish than a man chasin' his hat.

After much conversation with my husband about this symbol, I feel that the hat symbolizes home for Tom. "Home is where you hang your hat". Tom spends the entire film wandering between different "families" in search of a place to belong. Hence the dream of his hat blowing away from him. His hat is the only home he's got.

It's a symbol of death.

"Ever since Thucydides wrote his history, it has been on record that when the angel of death sounds his trumpet the pretenses of civilization are blown from men's heads into the mud like hats in a gust of wind." -- George Bernard Shaw, Heartbreak House

I've only seen the movie twice, but this is the impression I got from the hat. It represents composure and control. While Tom would never chase his hat through the woods, he does express a want for his hat every time he loses it. And he seems to lose it every time he loses control of a situation: someone ambushes him, he gets drunk and gambles it away, or he gets socked in the face.

The reason he wouldn't chase his hat is because he's very worried about the impression he gives others. He values dignity and the image of keeping a cool head even when he's in very hot water. If he has his hat, he's composed, and if he's composed, he's still got some control of the situation.

The theory, that the hat is a man's dignity is pretty damn good. Throughout the movie, as said, several times he loses his hat, it gets thrown away or knocked off his head. With the dream about his hat, how the woman goes on a totally different route, on saying the hat changed into something else, more beautiful. Tommy responds: "Naah, it stayed a hat, nothing more foolish than a man chasing his hat." This implies that no man chases his dignity. A man has dignity. He puts his hat on firmly during the end scene.

It always seemed to me that Tom (Gabriel Byrne) says 'hat' in a way that sounds a lot like 'heart.' "Nothing more foolish than a man chasing his heart" is a reasonable synopsis of the movie. Leo (Albert Finney's character) chases his heart trying to win Verna, and nearly loses everything for it. Tom spends the movie chasing his heart as well. He may be sincerely attracted to Verna, but he seems to realize that she may be using him. He also has deep loyalty to Leo. He takes risks for both. In the end his friendship with Leo wins out. Tom fixes things for Leo and keeps him safe. Tom is not sentimental about it, though, and sacrifices Bernie, thereby losing Verna. The risks he takes for Leo may be a bit foolish, but Tom is not so governed by chasing his heart that he fails to take more cold-hearted calculations into account.

To me, the hat symbolizes letting go of everything and being in the moment. In that moment, he doesn't chase the hat. He's out of hope, out of future, head full of static.

I always took the hat as a metaphor to a lid or cap on Tom's feelings and emotions. Like how much he does care for Verna and Leo, even though he acts like he doesn't. Verna even has his hat once, and Leo's people knock it off a few times with punches.

He is a very closed person, and never wants people to know his true intentions because he plays everyone.

The hat signifies pride, as in one of the seven deadly sins. Virtually every calamity in the movie stems from pride. Caspar's pride is stung in the opening scene where he accuses Tom and Leo of giving him the "high hat." That spurs the war. Bernie returns after being given a second chance because his pride is stung at the thought of Tom having seen him grovel. In the end, Leo puts on his hat, and Tom his, and they part, even though they love one another. Their pride covers up the meaning of their relationship and ultimately drives them apart, just as it drives Vera and Tom apart.

The hat is Leo. Tom is a completely cold-hearted guy, but if he loves anyone, it's Leo. And the movie is about how he comes to face that. Leo is like a father to him. Tom isn't exactly a paragon of self-awareness, either, and although he "knows all the angles," and negotiates a mind-bogglingly complex sequence of strategic moves throughout the movie, all of them in the interest of protecting Leo, he isn't necessarily conscious of his own motivations: "Do you always know why you do things, Leo?" In the final shot, we see Tom, from beneath the concealing brim of his own hat, watching the hat (with Leo under it) slip away from him through the woods, just as "it" did in the dream. He can't reach it, can't hold onto it, any more than he can hold onto Leo -- even after all he's done.

The hat represents Leo – Tom’s protection which covers his head.

Tom’s mind/intellect is how he survives. Leo protects Tom’s “head” without him we see Tom often roughed up.

In the dream the hat is blown off. Leo covering is removed. Verna is the wind.

Verna is the female equivalent of Tom. She influences Bernie, Mink to protect and advance herself. She could even have been the source of the idea to have Bernie sell the fixed fight information (increasing her resolve to protect Bernie). These qualities attract Tom. When she controls Leo, Tom’s protection is removed because Verna directs Leo into a detrimental situation with Casper. Instead of pursuing Leo to get his influence back and interfering with Verna directly. Tom realizes that Leo will continue to listen to Verna and his opinion "won’t count for something" anymore. Tom (a millar refining the circumstances surrounding him) crosses Leo. Continues the relationship with Verna and evolves to create his own protection.

The ending scene Leo is carried away by Verna. Tom puts his own hat on and pulls it down in a position so that a breeze will not pull it off.