OP: my dh also has it, and I have asthma (the conditions can be linked if chronic), so it runs in our family too!
The thing with eczema is that if you ask 10 people what works, they will give you 10 different answer. A PP mentioned tea tree oil and not giving daily baths. That would make my kids' eczema worse, especially the tea tree oil. A herbalist once prescribed a cream that contained some peppermint oil, and ds had the worst flare ever after that.
Every skin is different.
The main thing is to cut out any soap, and maybe go with a very mild laundry Detergent. We use surcare, no fabric softener etc.
Don't panic though. Lots of babies have a little bit of eczema at some point, it's their skin getting used to the environment. It will probably just go away again by itself.
Moisturise it as much as possible with a neutral cream (cetraben, aveeno, double base...). Cut her nails if she's scratching.
We have also had good experiences with the la Roche posay range of creams, especially their Toleriane range. It's very pricy though. But like I said, everybody will tell you about the ONE cream that worked for them.
good luck.