I've had eczema all my life and here's how I keep it under control:
-minimal water exposure, EXCEPTION: swimming pools with chlorine seem to help for some bizarre reason so swimming is good, long baths are bad
-let skin breathe daily: so let it air out for some amount of time every day without any products, then keep layering lotion on - i put lotion on every hour or two in the winter
-keep skin covered in cold & windy weather; cotton allows skin to breathe but protects skin from cratches
-keep nails short obviously
-Neosporin is a life saver on badly affected areas. I don't know if that's a baby-safe product, but I started using it when it come onto the market and I was in about 2nd grade at the time.
Honestly, being on the east coast during winter is so much harder on my skin than California weather. If you have any vacays coming up, take baby with you & let him bask in the warm sun. Good luck!