Please excuse the delay in responding to your post. I hope that not too much of what I have to say is redundant at this late date.
First let me say that your daughter is fortunate to have been born in the day of pediatric allergists and dermatologists. If her eczema doesn't clear up soon you should definately consult one or both kinds of doctors.
I have been struggling with eczema for over 50 years. Not all that much was known about the condition when I was growing up. The 'experts' of the day kept saying I'd 'grow out of it'. They were wrong. And in hindsight, often their best advice was wrong, too. Eczema has been my unwelcome lifelong companion and has left me with both physical and psychological scars.
The problem has always been that it is hard to figure out whether the outbreaks are being caused by contact with something in the environment or by something in the diet - or both. And it is confusing as well. For instance, I can EAT tomatoes, but I cannot tolerate the juice on my skin. I did eventually figure out that I am alleric to wool, but nobody mentioned that LANOLIN is a derivative of wool. It is found in many dry skin creams and lotions (and BABY WIPES). So when I thought I was doing something to help the situation better it was actually making it worse. Everything I read about sore nipples and breastfeeding recommended putting LANOLIN on the nipples. You don't even want to know where THAT took me..... And then there was the family doctor who said that my eczema was caused by 'stress'. He told me that if I divorced my husband the eczema would go away. Really.
I also learned some things about itching. First, I have come to believe that itching is the mildest form of pain. It is a constant nagging sensation that distracted me from work and play as a child and even interrupted my sleep sometimes. It made me very irritable.
The natural reaction to itching is scratching. Thus begins the itch/scratch cycle. The more you scratch, the more it itches - and it escalates. I can't tell you how many times I ended up with my hands in a bowl of ice water in the middle of the night just to try to stop the itching. As an adult I would sometimes pour alcohol on a raw patch of eczema knowing that it was going to hurt like hell, but then it would stop itching.
Some other things I learned about eczema by trial and error:
Sun and sea water help. At first the water made it sting, but then it felt better. I believe chlorinated pools helped too. (Check sunscreen ingredient label very carefully.)
Anything warm and moist (like the rubber gloves doctors were always telling me to use) makes it WORSE. Hot baths are NOT good for eczema. They make it itch MORE.
Alpha Keri bath oil in a WARM bath helped a lot. I would not recommend it for an infant though. They're slippery enough WITHOUT adding oil to the bath water. CAUTION: Bath oil makes the TUB slippery and the bather is more likely to slip and fall.
Aveeno Colloidal Oatmeal (another favorite of my doctors) wasn't particularly helpful. In addition, it smelled awful and the gloppy stuff was hell to clean out of the tub afterward.
The only thing that DID work (usually) was prescription strength cortisone ointment. Not cream - OINTMENT. Doctors are more cautious about prescribing cortisone these days, especially to treat a chronic condition over a long period of time. They are especially skittish about prescribing it for very young children.
Two words here: UNSCENTED everything (that touches the skin) Laundry detergent, softener, deodorant, lotions, makeup - EVERYTHING.
Avoid anything with alcohol. It BURNS eczema. Be especially careful with baby wipes on a sore bottom. There are few things that will make you feel like the worst mother ever more than wiping your baby's bottom and having her scream in pain. Trust me on that one.
Those are all the 'tips' I can think of right now. I hope they help. I don't want to see anybody go through what I went through as a child with eczema. Other children can be merciless and other parents can be ignorant or uninformed (read that "afraid their child may 'catch' something" from your child).
I'm sorry to say that my 4 year old granddaughter inherited eczema from me. A couple of weeks ago she came to play at my house. She had a rather large red spot of eczema on her cheek. A 2 year old from the neighborhood walked up to her, put her finger on my granddaughter's cheek, got a really REALLY sad look on her face and said, "Boo-boo". What a sweet empathetic child. I almost cried.
Feel free to contact me if you have questions about anything in this post. Good luck to you and your daughter.