My idea is that you start treating your son like an adult and trusting his judgement.
Pets provide something no medication in the world can - happiness and exercise. Happiness is at a terrible premium as you get sicker with CF, and even the best of us can get knocked down. Exercise is something we should all be getting to better our lungs and keep them in top shape; a dog encourages that.
As far as the "cleanliness" issues... if you are letting your house get filthy WITH a dog, you are probably going to let it get filthy WITHOUT a dog, too. In that case your problem is laziness, not a pet. Like a lot of people have said, a pet usually encourages people to be even more attentive to cleaning, sweeping, and dusting. Unless he has an allergy to dogs specifically (and I think he'd know that by now) he should be just fine.