J.,
A key question is, Why does my baby have eczema? Many pediatricians won't answer or will ignore it, and just treat the symptom. This happened to my daughter, and it took nearly 3 years to get an accurate diagnosis.
Eczema is a sign of allergies. Sensitive skin runs in my family, as do food allergies. There are different kids of allergic responses; the quick kind require a shot of epinephrine, the slower kinds manifest in other ways like digestive, skin, or sinus issues.
The eczema may eventually go away, but the allergy will not. If it's a contact allergy, she will get it each time she contacts whatever it is. If it's a food or inhalant allergy, over time her body's reaction will morph from eczema into ear infections, then inhalant allergies, then asthma. (Search "march to asthma" or "atopic march"). Per the article below, anywhere from 50% to 70% of kids with eczema can later end up with asthma. http://www.sciencedaily.com/releases/2009/05/090518213939...
Baby oil is a petroleum product--as in, made from oil. Also, many of the ingredients in commercial baby care products are irritants. (You can look up the safety of any product for free at www.cosmeticsdatabase.com.) I would recommend using a frangrance-free laundry detergent, and an olive oil bar soap. A little bit of almond, olive, or coconut oil as a moisturizer would be much safer than the medication. Shea butter is also very healing. You can get any of these at a health food store.
If you're nursing, your baby could be reacting to a food you're eating. Major culprits are wheat, eggs, and milk. If you're using formula, it could be an ingredient in the formula. Alimentum is considered one the gold standard for allergy concerns. If I'd had to use a formula, it would've been the only one my kids could've had.