You're playing poker. There is no "right" answer to which will work. There is, however, a right answer to preserving your professional reputation.
You accepted an offer from B. You now wish to renegotiate. You'll need to be very forthright. Simply admit that you were unaware of your market value, and that it would be irresponsible of you to not accept A's counter offer for you and your family. You would be willing to move forward with B, but they would have to offer $x. Don't give specifics about A's offer. Just state what you need.
Be aware that A will likely pass you by for any raises or extra remuneration for the next raise cycle or 3. So if these are long-term plays, look past 10% to your bigger opportunities. If you plan on shifting jobs, take what you can get where you find it.