Family wealth is an indication of how adept a particular family is at hoarding their gold. The wealth is accumulated by one or more characters sending their gold home instead of spending it in their realm.
As your family wealth increases, so too will your ability to support new and existing characters. Extant members will be able to draw upon this wealth in times of need, at the expense of incurring a natural prestige loss due to such a desperate action. Family members at the beginnings of their careers will also receive larger amounts at the start to help them along their way.