It’s been mentioned that alcohol is a diuretic, so drinking water helps with the majority of hangover symptoms, and it breaks down the store of glycogen in the liver (glycogen -> glucose, which goes directly to the liver): Frequent urination will also expel salts and potassium resulting in headaches, and nausea.
I know that’s not really helping, so here’s some things that work (and a short explanation):
Fruit Juice: Good old fructose will help you get your body’s energy back up, and it tends to have some wonderful vitamins and nutrients (and anything with C or B vitamins will help).
Bananas: They have a lot of potassium which as I said is depleted, so they (along with anything with a lot of potassium) will help.
Eggs: Eating anything will replenish lost electrolytes, but eggs contain cysteine which breaks down the toxin that essentially causes a hangover (acetaldehyde for those interested).
Aspirin: As other people have been said, this is found in willow bark in the real world, and it does help (I’ll forgo the reason) so go with it (Note that if you’re thinking about this in real life, people with sensitive stomachs should avoid aspirin like the plague...it can make the pain worse).
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Now that all that is said, if you’re DM is fine with it, I’d feel free to make up (and document) any number of herbs and other substances to cure a hangover. Since it should be a fairly easy heal check for someone with 15 ranks in the skill (or so I would think), the rest is flavor, so go nuts!