I've read through this thread and just wanted to add that Yes There generally are things that 'Trigger' Eczema. My sister had it up until a few years ago (she's 35) and one of My aunts had it so bad her fingernails looked like waves in the ocean and she wore fake nails for most of her life.
Things like Eczema and asthma tend to be hyper allergic reactions to environment and or foods.
I see a lot of people telling you what to 'add' to his diet, but I really think you need to find the source and cut it out. Maybe you need to go see a naturopathic doctor to help get to the bottom of what your son's triggers are.
Generally gluten and dairy are the fallback culprits, but for my sister she found that acidic foods like Tomatoes were also a trigger in addition to dairy. But it could be something as unknown as the chemicals in the carpet in your house!
Testing for food allergies with the conventional scratch tests is not the easiest to administer on a 2 year old. My naturopath has a vega test that he uses which had a machine with a pen that is touched to the finger and is probably easier and less traumatic on a 2 year old.
I wonder how early was it that you introduced foods to your child? especially things like eggs, dairy, wheat. And did you introduce one at a time and test his reactions?
My son was completely disinterested in food of any sort until after 1 year (to the point where he wouldn't open his mouth to a spoonful of anything hungry or no) and is really only interested in any variety recently at almost 2 1/2. Eaten his first mouthful of egg in the past week even though it's been offered periodically.
Incidentally he is sensitive to raw egg whites, and has had pretty severe reactions to 'Unknown triggers' a few time since he's started eating foods. I think babies somehow instinctively know.
Anyway, I think you need to start 'removing' things one at a time and observing the difference for 2 weeks at least. I know removing milk can be a big issue with calcium, but I believe that Yogurt can be kept in as it has beneficial bacteria which can help to compensate. Also I find grated broccoli or pureed spinach in things can get it into them without them knowing (ie. On pizza, in sauces etc)
I think it was brought up by someone else, but I would also recommend not using soap or anything to wash him on a regular basis. My son bathes almost daily because it is part of his bedtime routine, but we use soap or shampoo maybe once every 2 weeks. And then only something natural - we've found a calendula shampoo/body wash by Weleda that is really nice to his skin and scalp (he had cradle cap for the first 1 1/2 years). We had used the coconut oil for a moisturizer for a while, but found that he got pimples from it on his legs or torso or arms and now we just use a glaxal base though I'm sure others would take issue with it but it is pretty benign.
Whatever you do and wherever you find the answer I wish you all the best, and a speedy resolution