Can mild cat allergy cause eczema?

Hi and thank you to anyone who may be reading this! Thanks for taking the time to read it, and I'd love to get some other people's opinions just to bounce this off of.  
My son, who is turning 5 in just a few days, has had eczema since he was 3 months old.  At first the Dr's thought it was just your run-of-the-mill infantile eczema, but seeing as how it never went away, they are now searching for causes and solutions (which is hard at the moment because we don't have insurance). Anyway, we were able to see an allergy specialist, and they did the scratch test on his back. The only thing he tested positive for was a cat allergy, and apparently it was such a "mild positive" that they had to retest to confirm. Obviously it was confirmed, but I thought pet allergies were more itchy eyes, sneezing, ect, which my son has none of (we have 2 cats at home). When I asked the Dr if the cat allergy could be causing his eczema, I didn't really get an answer. It was kind of a "hmmm, maybe...." type thing.
So I was just wondering if maybe anyone else has had this kind of reaction to a pet allergy? I would just hate to have to get rid of the kitties, but my son comes first- no question!
Just trying to find the next step. My poor son scratches till he bleeds, and he's never been fully healed since he was diagnosed- he's always had a spot of it on him somewhere, and most of the time it's his whole stomach and legs! Poor little guy, no matter what kind of gunk we spread on him, it just doesn't seem to take it away. Sometimes it makes it more manageable, but more often than not it's out of control. He's sick to death of scratching and bleeding and being in pain, not to mention having to take so much time out of every day to put whatever "miracle" cream or rx on him that we've been told to try this time!
So if anyone has ever heard of having a skin reaction to a pet allergy, I'd be grateful to know! I want to find the best home possible for my kitties, and I'm sure that will take some time.
Thanks again!