Hi D.,
My son has had eczema since birth. He's 6 now. It gets worse in the spring and summer because of his allergy to pollen, as well as in the winter because of the dryness. I took him to couple of dermatologists, they gave him prescription topical cream (stronger version of cortizone). I only apply it when the condition gets very bad. Otherwise I just put lots of lotion on him. Since your son's condition is mild and he's not itchy, just put on lotion frequently, like 4 times a day. You can use any unscented lotion like Aveena, Eucerin, and Centaphil. But if his skin gets really dry, these lotion may not help, then you can use the cream version of those brands (I've tried all of them. Eucerin is the most oily one, but with my son, it worked the best). The cream has really worked for my son these days since he goes to school and it's impossible for me to apply lotion several times a day for him. So I do it once in the morning, and once in the evening after shower.
Good luck.