What can I do about my 4 month old baby's eczema?

See an Allergist. Eczema in infancy is often associated with a food allergy. Even when it is not, your allergist can give you excellent advice on gentle skin care, moisturizing, and appropriate medications. If there is a food allergy, avoidance may dramatically improve the rash.
Avoid soap. #1 before reaching for Hydrocortisone or rx meds is to avoid soap. Babies that age do not need soap which will dry skin out further. As soon as he/she is done bathing, generously apply vanicream moisturizer 1-2x daily all over. Otc Hydrocortisone can be applied to active eczema under the moisturizer and if not improved in 2 weeks, see a dermatologist for eval and to rule out infection.