Gamexplain recently released a Mario Odyssey analysis in which they speculated that Mario could switch hats and get different abilities, so i'm here to speculate more on what these abilities could be.
American Football Helmet: This one is actually shown outside a crazy cap door, so it's very likely this one is in the game. Most likely lets Mario charge into enemies for a kill.
Top Hat: Also shown outside crazy cap. Since it makes you taller, could make you jump higher. Or since in real life a top hat is a wealthy person's hat, or at least was, could have something to do with getting more coins, or those purple things we've seen. I also believe those purple things will be a currency for buying the hats at crazy cap.
Beanie/bobble hat: Protection in cold, or perhaps more control on ice. Would have something to do with ice levels.
Straw hat: Protection in heat
Hardhat: Gives you protection from things falling on your head, like wearing a buzzy beetle shell in mario maker. (by extension, you'll probably be able to wear a spiny shell like in mario maker too, which would be a better version of the hardhat)
Chef's hat: We saw mario holding turnips, so maybe with this hat he can eat them and other things for certain buffs?
Fez: Does nothing, but you'll wear it anyway because it's a fez. What kind of person wouldn't wear the fez?
Anyway, those are my ideas – hope you enjoyed. Does anyone else have more?