Eczema can be an internal yeast problem. I would detox as much as possible and de-yeast as well. My daughter was born with thrush in the mouth and systemic yeast, it was coming out her bum. A few months later the doc said she had the worst case of eczema he had seen. She doesn't have any now...we de yeasted her. Dandilion extract can detox the liver and I would start there. Caprylic acid is also great, as is raw virgin coconut oil, on the skin and internally. Also, 1 T a day raw virgin cold pressed flax seed oil (found in dark bottles the fridge section of good markets) brings oxygen to the cells and helps clear toxins.
ps... are u sure its eczema? I had a rashy thing like that on my eyelids once, turns out my makeup- my eye shadow- was infected with bacteria. Throw out any cosmetics and don't use any for a while. Use the coconut oil on the lids a couple times a day.