My daughter has eczema too and there could be several factors to this. Yes, it could be the food. The earlier you start food, (earlier than 6 months) the more likelihood you might have a reaction to certain foods. She could be allergic to certain foods and that might pass with time, but she could also have sensitive skin. What are you washing her clothes in? I wash both of my kids clothes together and use dye free, perfume free, all clear detergent. ALL has a great new product out and her dermatologist suggested I use that for their clothes. She has occasional breakouts and I have finally gone with a much stronger steroid creme to help heal the flare ups.
Food can cause a reaction and its' good to stop the food as soon as you see the reaction and then try again with it or go on to something else. Cereal is best to start off with first, then greens vegetables and introduce one food at a time repeating it over the course of several days to see if there is a reaction. Check out: www.http://www.wholesomebabyfood.com/ for awesome tips and advice to what you should give and some great recipes too.
My kids were not allergic to any foods, but they reacted to the soaps we were using. We only give baths every other day, rarely to we do bubbles, b/c that dries their skin out. As soon as they are out of the tub, barely even towel try, I grab the tub of Aquafor (sp?) and go to town on my daughter. She has the worst out of the two. The Aquafor will keep the moisture from the water in her skin instead of drying it out. Then I follow with her prescription creme if she has any problem spots. As far as soaps. We use all clear, free of dyes and perfume hand soap (Method has a great one available at Target or online) and ALL (brand name) detergent for their clothes. I try to avoid the hand sanitizer b/c of the alcohol, but do what you can. It could very well be an allergic reaction to the foods. If it persist you can go see a dermatologist for a prescription creme to use and very helpful advice on what to use daily.
Good Luck!
A. B