The people who post on the allergies thread (within Health) on here are a wealth of information.
Sorry mousymouse but the advice you had about food isn't really correct. It's true that excema can be caused by many things other than diet, but diet can be a major factor for many people of all ages.
Depending on how severe the eczema is you may want to try and get a referral to a specialist but you probably won't get this until after you start weaning (if my health authority is anything to go by!). Based on my limited experience and (6 months old, severe eczema, has been tested positive for egg, nut and cow's milk protein allergies) and lots of conversations with specialists I would suggest avoiding those foods which are commonly allergenic (e.g. eggs, nuts, cow's milk - inc cheese, yogurt etc, soya) in the first months of weaning. This includes coked eggs e.g. in cakes etc. I have also been advised to avoid gluten at the beginning to be on the safe side. Use the 3 or 4 day rule when introducing new foods, keep a food diary so you can spot any patterns in case of reactions and when you want to introduce any of those more allergenic foods take extra care. Root veg and those things usually described as 'first tastes' are generally pretty safe, but some fruits (particularly seeded fruits like strawberries) can cause a reaction in allergic children. You may want to discuss with your GP whether your child might need any vitamin supplements to ensure they're not missing out on anything if you're restricting the diet at all.
If you are still breastfeeding, be aware that allergens can pass through your milk and you might find that cutting out some things may improve your child's eczema. I was really cynical about this but the change in my child's skin since I excluded eggs/nuts/dairy from my diet is extraordinary.
It may sound daunting but I've found it much easier than I expected. Good luck and enjoy!