Its hard to say for certain what this is but chances are its some sort of allergy or eczema. You could apply a cortisone cream, an anti-itch lotion like Sarna, or combine them. Sometimes holding a cool, wet wash cloth provides temporary relief. An antihistamine like Zyrtec children's syrup might also help.
None of the above will help if the problems are caused by a skin infection. Some fungal infections could also cause these symptoms. Topical anti-fungal creams could be tried if the above doesn't work. The ingredients in some of the fungal creams are toxic if ingested, so she would need to keep her hands out of her mouth if antifungal cream were used.
I forgot to mention scabies as a possibility. Not a likely one since it hasn't spread beyond the hands in 5 days. Scabies would typically spread to other areas.
