M.,
I just read the many responses and am surprised that so few moms have been advised by their pediatricians to check for ALLERGIES as the cause of the eczema. I have a 23 year old daughter who began exhibiting signs of eczema at about 6 months. We went to dermatologist after dermatologist with little luck, until finally someone told us that allergies, eczema (and asthma too) are "related," and we needed to figure out if she had allergies to get control over the eczema! If allergies are the root cause in your situation, you may be able to eliminate outbreaks by identifying the allergens and elimating them as best you can.
In our case, it wasn't anything she TOUCHED. Food allergies were the main culprit - we didn't even know she had food allergies! We had to start over, introducing one food at a time to see the reaction. It turned out she is allergic to dairy, eggs, most nuts (especially peanuts though) and citrus. All food with dairy or eggs had to be removed from her diet, and we had to learn to read ingredients lists, looking for pseudonyms for diary (like lactose and caseine). Once we did, though, eczema was gone and so was a lot of cranky behavior.
Other triggers: laundry detergent (buy the hypoallergenic or "FREE" versions), soap & shampoos (use Aveeno, Dove or one of the other products suggested here), and believe it or not, weather. Cold weather chapped skin became eczema for our daughter. Hot weather heat rash too. There's not much you can do about this, other than stay cool (literally and figuratively) and be liberal with the body lotions and creams after and between baths.
A couple of last-resort things when eczema just won't heal: (1) put preferred cream or ointment on eczema and then wrap the area in saran wrap and tape it in place. Leave over night if your child will tolerate it. This is called occlusal and it forces cream deeply into the skin to facilitate healing. Obviously do not do this on the face. (2) Anti-biotics. Once in awhile, the broken skin hosts some sort of staff infection, and needs antibiotics to clear up. (3) In a really no-win scenario, we've resorted to a very short course (one week) of oral steroids. I tried hard to avoid that, but over the years, we've done it five or six times. There is a blood test for steroid levels, and I was paranoid enough to have it administered to my daughter - apparently it takes a lot more steroids than she had to have a negative impact.
You might also be interested in the web site run by the National Eczema Association. I joined (free) to get a monthly newsletter with the latest medical and non-medical treatment information. Here's a link to their kids' info. http://www.nationaleczema.org/lwe/children.html
I hope some of this helps. With any luck, your child will outgrow this in a few years. Many do. Feel free to contact me personally if you want some been-there, done-that support. My email is ____@____.com.