My older son went through a bout of very severe eczema. Yeah, it itches a lot (he'd scratch till he was bloody, poor guy).
I'll second what Caring Touch and TurboClaudia said... there is good documented evidence supporting both of these. However I'll add that eczema is not really well understood, and food may not be the answer (but if I were you, I'd hope that it was, because it's just about the easiest answer).
If you have access to a pediatric allergist, I'd see her/him instead of a regular pediatrician. In my experience, the generalist (the ped) doesn't always have all the info that can help. A lot of what we have learned which has helped a lot we got from the allergist, and a lot of it has nothing to do with drugs (eg, we learned to give our son frequent tepid baths and then "seal" him with something greasy immediately after the bath... anything you can do to keep the skin gently moisturized and not irritated helps). The biggest risk of hyrdocortizone is that overuse/prolonged use can cause thinning of the skin (which isn't going to help the eczema situation).
Some other things you might think about... eliminate any personal care items (shampoo, lotion, detergent, soap) which have any dyes or perfumes. Double rinse your laundry. Avoid sunburn and overheating (they can exacerbate the eczema). Go very, very slowly with introduction of solids, and stay away from foods which are common allergens (wheat, egg, dairy, soy, nuts, corn, berries, etc.).
Good luck!