How does the family background mentioned in the statement of purpose affect a PhD application for admission in U.S.?
How a statement of purpose is read varies from department to department, and from person to person.  That said, I think for most people reading your statement, it wouldn't matter much.  Yes, many people are for socio-economic diversity in admissions, so scenario B could be slightly helpful, but really this might only make a difference in very borderline cases.  (Of course, you should be truthful, and not claim to be Pete Clark's offspring if you're not.)

What's important about the statement is that we get some sense of your motivation and goals, and possibly mathematical maturity.  It's also good if your expectations about grad school sound realistic.  By the time you are applying for PhD programs, between your academic history and your letters of recommendation and a statement about your motivation/goals, we should have a good sense of your likelihood of success in our program without any information about how you grew up.  We care more about where you are now rather than how you got there, so to speak.

My advice is, if you feel it's useful to explain where you came from to explain your motivation/goals, then by all means feel free to.  If not, there's no need to bring it up.