Hi, S.!
Eczema should not be the cause of "red dots" on your baby's face, and obviously, it's not a "birthmark", because your daughter is a year old and is just now developing the problem. Is there an optometrist or ophthalmalogist in town that would see your daughter? And did your pediatrician rule out chicken pox, measles, and mumps, among other conditions? (I actually had chicken pox TWICE....the 1st time I was 4, and the 2nd time I was 7 ~ still have "pox" scars on my face, even though I didn't scratch them.) It may also be an allergic reaction to something...have you started using anything new with her, like bath products, or laundry soap, or even new foods this week?
My son has eczema, but we use Renew Lotion and he very rarely has an outbreak of it anymore! (Actually, the only time he does have an outbreak is because we haven't used the lotion in a while. He will actually get his lotion and either get me to put it on him, or try to put it on himself! He is sooo cute when he does that!) Let me know if you'd like to know more about it.
Hope you and your daughter get to feeling better soon! Let me know what you find out...my address is ____@____.com.
~J.~