My players just got stuck in a Feywild forest and might be on the plane for a while. In Irish myth, fey society is basically a feudal one, if I understand. To me, this means it will be similar to the structure of the home Material Plane country, and adventuring there might be not very mystical. How might I change it up?

Most D&D lore emphasizes that fey are mysterious, enigmatic, even alien, creatures, so I'd avoid making anything seem too similar to a kingdom on the material plane. It's true that fey have kings and queens, vassals, courts, and other such things, but beyond these surface-level aspects of the feudal system, fey society is supposed to be strange.

Play up these aspects of the fey, by portraying them as mischievous and a little beyond understanding. They shouldn't fit any stereotypes of good and evil or of any other moral structures used by the civilized races. One good way to do this is to think of the Feywild like a dream. Aspects of the fey realm may seem like analogues to ordinary things in the material plane, but they don't obey the same principles; or alternatively, things there may seem extraordinary, but their purpose may turn out to be rather ordinary after all.

Weather, illumination, and other environmental effects may change on a whim or as a result of the changing emotions of powerful fey beings. The practices and etiquette of fey courts may seem at first similar to their human equivalents but then quickly devolve into the nonsensical. And no matter how ridiculous something may seem to the PCs, a fey is just as likely to find it humorous as they are deadly serious.

This article has some more useful tips.

Thanks!

Fey society is feudal, it is also very much a foreign thing. A Midsummer Night's Dream, if you familiarize yourself with that to some degree, has wonderful ideas to pull from. As jreggers points out, it is very much about putting everything on it's head and then painting all the blues with yellow. However, it is not quite so odd as to be the Mad Hatter's tea party, just...different from the way we are familiar with things.

Some of it is, though.  Wonderland itself isn't all that different from a subsection of Feywild, where thoughts, concepts, and things are infused with the primal power of life and either animated or given an incarnate representation.

I have a stereotype.  I always, but ALWAYS, play something fey.  Either the character is fey directly, or has some direct connection to the feywild.  Why?  Because fey myth has capticated me.  While the D&D representation is strong based on Celtic myth, the celts were not the only source of natural spirits that eldritch appearances or behaviors; i personally have always grouped yokai from japanese myth in with fey, as their overall tendencies are identical. 

A video that may help: TMG Ecologies: the Feywild

I'd also recommend The Dresden Files as a series of books with good examples of Fey which are feudalistic, but very wild, different, and alien.

Plus I think they're a pretty enjoyable read, and a pretty novel twist on fantasy magic, Fey aside.

Summer Knight is the first introduction of the Fey courts, a pretty good source of the kind of material you're looking for, and if you want to read just one of them, go with that, but for spoiler reasons, the Fey Courts feature prominently in the latter third of the series.