Topical creams are of limited use. A lot of people use steroids but of course they can only be used for a short period of time, and you don't want him to get his hands in it (or any of the creams, actually). But you're only treating the surface of the skin, and you have to recognize that eczema is coming from much deeper inside. It's a systemic issue, and because the skin is the largest organ, the eczema can show up anywhere.
So treating it medically is of limited effectiveness for many people, and your son seems to be one of those. Food science is the other approach - most of the people I work with in this field give a specific children's supplement (which has many other benefits as well because it is formulated by pediatric nutrition scientists). We've had some really resistant cases that don't entirely respond, but those usually resolve with the addition of a unique fiber product (soluble and insoluble combined, which is hard to find) and which detoxifies. The basis for the science is that there is a nutritional imbalance somewhere, either with intake (common with kids especially) or with absorption, so the imbalance results in toxins which are really pushing out through pores, mucus membranes or digestive elimination (feces mostly, plus urine).
You could go through a long litany of tests and start eliminating foods, but it's a nightmare to maintain and it totally affects your lifestyle, including going to other people's houses and to restaurants. The vast majority of food "allergies" (other than hard core anaphylactic reactions) can be significantly reduced or eliminated, not by eliminating, but by adding what's needed to process the offending nutrients. Even highly allergic people can often get to the point of consuming the trigger food, but if they want to avoid it no matter what, at least the food science approach can get them to the point where they don't have a life-threatening reaction if accidentally exposed (through cross contamination, for example).
With all the new work done in the past 15 years with repair of the epigenome (the mechanism that controls gene expression and how cells specialize by genes turning on and off in different cells), there is little reason to continue to suffer. Cellular nutrition has been around for 30 years but most doctors haven't had a single course in nutrition so they aren't up to speed. There emphasis is on treating the symptom rather than preventing the problem. That approach is great (and appropriate) for acute care, but it's ineffective or laden with side effects in many cases of chronic conditions.
I never recommend that people start extensive elimination diets and unpleasant testing, when a much more effective approach is available. You can still treat eczema and other conditions short term with the prescription creams while the body balances.