J.B.
As the others have mentioned, Benadryl is the first line of defense for an allergic reaction of any kind as its shuts down the histamine reaction. Keep both liquid (oral) and topical cream in your medicine cabinet.
Eggplant is a food that's not likely to cause an allergic reaction so when it does, it's usually a topical or oral food allergy, which is different from a regular food allergy. The body is basically reacting to a protein in the food as if it were a pollen, showing signs of surface-level irritation (it feels like tingling in the mouth or tongue, or like you're tongue is wearing a sweater) but not anaphylaxis, which is the deadly reaction more commonly associated with true food allergies . Eggplant oral food allergy is more common in people of Indian descent but it can happen to anyone. Oral food allergies are usually associated across whole families of foods and are linked to severe pollen allergies. My oldest son has had severe allergies to most tree pollen in the spring for 10 years and a couple of years ago, started to develop oral food allergies to raw fruits and vegetables and the list of reactive foods is growing and now includes some nuts. For most produce that causes oral/topical allergies, cooking the food changes the protein that the body thinks is pollen into a form that the body does not confuse with pollen, so many people with oral allergies can eat cooked or processed foods, just not raw. Eggplant is one of the foods where one of the reactive proteins retains it's pollen-like structure even after cooking, as do some nuts.
Anyway...give Benadryl for now and you should see the rash clear up and know that he's fine. Obviously avoid eggplant for now and talk to your pediatrician. I would imagine that with a baby so young, it would be prudent to have allergy testing done so that you know what you're dealing with. Maybe it's just a freak thing that he'll outgrow later, but if there are other foods in the eggplant family that would cause a similar reaction, it would be good to know that now, and if this is a true oral food allergy, you're probably going to be dealing with pollen allergies as well. Other nightshade vegetables are tomatoes and peppers and there are probably others in that family that should be avoided for now.