"The Lottery" plays around with the concept of family in interesting ways. The thing is, each person in the lottery must draw by household, so this is the moment, each year, when belonging to a given family has the most socially recognized significance. In the midst of this collective ritual, though, it's during the lottery that the emotional bonds that connect mother to child, husband to wife, and friend to friend, become completely insignificant. Once the lottery has ended, family bonds reassert their importance, and the families who have lost members mourn them. So Jackson is clearly drawing a line between the social place of families (with their male heads of households and unfair distribution of luck) and the emotional importance of family ties, which is a private matter.
"The Lottery" distinguishes between family as an emotional bond and family (or the household) as a social bond.
The household in "The Lottery" is a microcosm of the village's overall social organization, with women deferring to men, and children deferring to their parents.