I do disagree that a pet is not just a pet...it is part of your family, hence why this decision is probably very difficult for you.
My husband is allergic to cats too...but by experiment I found he is actually allergic to the litter, not the cat itself. He would sneeze, his eyes got so red I don't really know if he could see or not...(I now use the tidy cat, the stuff that clumps) I had to move it out of our living area and into the basement...far away from 'humans'. I still have a cat who gets upset with me and will go and mark in spots...I have to clean his TWO litter boxes everyday to make him not do this to my carpet.
If you have tried everything under the sun you can think of, the problem is persisting or getting worse, re-homing them would be for the best...as hard as it may be.
There is also the option of trading in the cat for maybe a small dog, or a family fish tank that everyone could help put together...the kids will heal, and yes they will be sad that is a given...but overall you have to do what is the best for your son.