As always, Dr. Campbell has a super answer. If you were to carry a first aid kit, that's what you'd want in it (give or take). I have to admit that I don't carry a first aid kit myself, and I'll give you my rationale.
I probably do most of my diving from day boats on vacation in populated areas, on liveaboards, or as local shore dives close to home. A big first aid kit is too bulky and heavy to take on the plane and then lug onto the boat. If you sustain a minor injury on the boat, it will usually wait until you get back to town. In the unlikely event you have a major injury, you probably won't have what you need to take care of it in a first aid kit anyway. Liveaboards generally have stuff on board to hold you until the helicopter gets there (paper towels and duct tape will do in a pinch), but I do carry a supply of bandaids because liveaboards tend to run out. Shore diving locally is where a first aid kit would most likely be of use, particularly if you dive in more remote areas. It would be easy to carry in the car, and if you happened to have an injury, you wouldn't get so much blood on the car.
I do carry a supply of over-the-counter (OTC) "comfort" medications on trips. It's easy to keep all of the below in a small, light, easy to pack kit.
When diving in salt water I keep a couple of small spray bottles (from The Container Store) of vinegar in my dive bag for hydroid (jellyfish, coral, etc.) stings. A spray is easier to apply evenly than pouring, so if you have a spray bottle you can carry a smaller supply. This I take on the boat. The rest will be back at the hotel.
I carry the non-aspirin pain reliever that Dr. Campbell mentioned (eg Tylenol) for those aches and fevers and ibuprofen (eg Nuprin, Advil, etc.) for aches, fever, and inflammation. Naproxen (eg Aleve) is in the ibuprofen/aspirn family and is a good choice too.
The cortisone cream he mentioned is a must-have for any dive trip. It will help relieve the symptoms from hydroids and bug bites and stings. It is also good for those mysterious rashes that show up sometimes or anything that itches.
Diphenhydramine (eg Benedryl) antihistamine pills and cream are good for allergic reactions.
Meclizine (eg Dramamine Less Drowsy, Bonine) is my OTC med of choice for seasickness. It is also in the antihistamine family and will do in an emergency if you don't have Benedryl.
Pepto-Bismol is the OTC drug of choice for traveler's diarrhea IMHO. It not only physically slows the diarrhea it also absorbs some of the nastier bacterias' toxins to keep them from making you so sick. It can also be taken as a preventative to keep you from getting traveler's diarrhea on your trip. Immodium will also stop the diarrhea, but sometimes works so well that you'll stop for a week. It also does nothing about the toxins from the bad bacteria and can actually make an illness worse in some cirumstances.
A 12 hour formulation of pseudoephedrine (eg 12-Hour Sudafed) is my pick for sinus and ear congestion, to help with equalization, and to ease the symptoms from sinus and ear barotrauma.
A 12 hour oxymetazoline nasal spray (eg Afrin) can also help with congestion and equalization. Shorter acting decongestants wear off too fast IMHO.
An antacid type medicine is handy to have around for all those strange foods you eat on vacation. They can also help settle the stomach sometimes if the meclizine doesn't work. Pepto-Bismol will do, but tends to be constipating. Liquid antacids are sometimes messy, and antacid tablets are too short acting. I like to keep an OTC medicine in the H-2 blocker family in my kit. Some examples would be Tagamet, Zantac, Pepcid, and Axid. Pepcid and Axid are longer acting. (Can you tell that I like longer acting meds for most applications?)
Eardrops for swimmer's ear are good to have around if you tend to get irritation of your external ear canals from diving. You can make your own (formulas are here on the Scuba Board site in other threads and on Dr. Campbell's web site). Otic Domeboro is my preferred eardrop, but it is by prescription only in the US. Star Otic is an OTC eardrop that closely approximates Otic Domeboro, and I think the best if you want to buy it at the store.
I keep a couple of prescription medicines in my kit too. I keep Transderm-Scop patches as my Rx drug of choice for seasickness, and I keep an antibiotic appropriate for the really bad cases of traveler's diarrhea. A prescription anti-malarial might be appropriate for some dive trips.
This isn't an all inclusive list. I probably left something out.
All of the above can help make a dive trip more pleasant or even salvage a trip. But keep in mind that any medicine you take hoping it will have an effect can also have a side effect- whether it's by prescription, OTC, or "all natural". I could come up with a pretty good scenario for each of the above medicines where they should NOT be used when diving. Before you take any medicine you should make yourself aware of what's in it, what it's supposed to do for you, what it can do to you, and which individuals should not take it. Of course any prescription medicine should be taken only with the advice of a physician.
HTH,
Bill