Make an appt with a dermatologist.
Hello,
My daughter had/has that and it started at about the age. The dermatologist said it is "eczema of the finger tips". He actually had a long Latin name for it, but that is the translation. Her toes and feet peel now as well. Hers cleared up during the summer months when she was outside a lot, but come fall and being inside more it would come right back.
He should see a dermatologist or allergist for an accurate diagnosis and treatment. Since he is sick a lot, I would recommend an allergist to treat all his allergic conditions. I hope he gets to feeling better.