What is the importance of family in "Good Country People"?
The importance of family in O'Connor's "Good Country People" is not central to the story, although family does play a part.
Mrs. Freeman (which she isn't--free, that is: she's a tenant farmer) and Mrs. Hopewell (she can hope all she wants) are simplistic and simple-minded. They live by simplistic platitudes and cliches. Joy/Hulga thinks she's superior to everyone else and lives in a constant state of antagonism with her mother. Mrs. Hopewell considers Mrs. Freeman's daughters "two of the finest girls she knew" because one is eighteen and has many admirers, and the other is fifteen and pregnant--low standards for being the "finest." The mother is far too simplistic to ever understand her daughter.
But, again, revealing the dysfunctionality of families is not central to the story. The narrative is predominately about intellectual pretensions.
Hulga thinks she has a monopoly on the knowledge that everything is meaningless (nihilism). She thinks that her great learning has led her to this understanding, and other characters do not possess this understanding.
But the intellectual are not the only ones who understand this. Manley Pointer (sexual connotations aside) also knows this "truth":
"...And I'll tell you another thing, Hulga," he said, using the name as if he didn't think much of it, "you ain't so smart. I been believing in nothing ever since I was born!"
The poor and uneducated can also know the "truth" of nothingness.
Ironically, Pointer's stealing of Hulga's artificial leg reveals the opposite of nothingness--it reveals the presence of evil. The implication is that if evil exists, then good must exist. Also ironic is the fact that Hulga goes to the barn to seduce Manley, and is seduced by him, and in the process reveals that she, too, thinks in the same terms as her mother: she, too, thinks that he's just "good country people." And she is just as incorrect as her mother, of course.
One aspect of family that is important appears in the way Hulga interacts with her mother. Because she has an education, she feels that she is far superior to the "country folk" family from which she has been born and raised. Hulga was born "joy" into the "Hopewell" family - the names seem significant as well, hope and joy. She rebels against her identity as a part of this simple family by changing her name. This intellectual superiority, coupled with a disability (the wooden leg) sets her apart from the family. She believes that her intellect is better than "good country people" values that she assumes anyone she comes in contact with will possess. Because her family possesses them, she is easily manipulated by Manly's seduction.
Ultimately, when she is left alone by Manley, we can assume that her family will become significant again as she must wait for them to rescue her. Without her glasses she is blind (she was already to an extent, but more in a common sense way!) and without her leg she cannot get anywhere. So, she is left with the isolation from family that she sought when she rebelled against them, but also with the need for family to come to her aid.