My DD was also diagnosed with eczema at about three months, and we also do not have it in the family.
Was it aqueous cream that the GP prescribed? We were given this initially and found it useless.
The main way that we have managed it is by cutting out all soap products, and also bubble baths and creams and anything really. You can bathe him in just plain water, or we use Cetaphil cleanser but there are other non-soap products which might be okay. Many pharmacies have free samples of Cetaphil so definitely worth trying if you can get a free trial. Then after bathing, make sure you dry him thoroughly, particularly in his creases, backs of knees, etc, and then liberally use a moisturiser for sensitive skin (we also use a Cetaphil one). Don't save the moisturiser just for after baths - use it at other times of the day too if he needs relief.
We also find that Papaw ointment can work well when we need instant relief. Just be aware that it kind of sits on the skin as opposed to being absorbed, so when your son is a little older and more mobile then it will be rubbing off all over the furniture and carpets.
While you need to be conscious of the effect of sun on baby's skin, you should also be aware that fresh air can be really good for eczema, so do expose his skin to sunlight when you can. Similarly try to dry laundry in fresh air and sunlight, although we've found that what kind of laundry detergent we use doesn't make a difference.
I breast-fed DD up to 12 months and during this time repeatedly made modifications to my diet but nothing made a difference, so I don't believe that my DD's eczema is/was caused by diet.
Ultimately, my advice is that there are loads of products out there and some will work for some people but not others, so you've just got to try anything until you find what works for you. But it can get expensive if you're out buying a new cream every week, so don't be afraid to ask, if a friend recommends something, if you can borrow a bit of theirs to try.
Hopefully, you will soon get to the point that we're at where it is manageable. It isn't problem solved, but DD isn't too irritated and you can't see too much. But it's such a fine line - she only has to skip a bath one night, for example, and she'll have a major flare-up which will take days to bring back into line. I have heard that children can and do grow out of eczema, especially those who have no family history of it, so fingers crossed that that is what will happen and it won't be a life-long affliction.