Whether of not the son had been diagnosed with any mental problems (the information Shockwave mentioned isn't really reliable) is still a matter of debate, but one would be limiting gun access to those with severe mental problems living in their household.
I think guns should be treated like driver's licenses: easy to get, you need a permit, there should be some training (at least for handguns). So, I can see an argument for ensuring people are competent before they can access guns. But regulating the household of a person with no mental illnesses? Sounds pretty tough to enforce.
Also, what's the difference between a person with severe mental problems and a young kid living in the house?