When at last she slept, she dreamed of home. The kingsroad wound its way past Winterfell on its way to the Wall, and Yoren had promised he'd leave her there with no one any wiser about who she'd been. She yearned to see her mother again, and Robb and Bran and Rickon… but it was Jon Snow she thought of most. (2.Arya.44)
Arya's entire story is centered on two conflicts: First, she needs to survive; second, she wants to return to her family. For Arya, family is not just a source of comfort and support. It's also a means of survival since she's, like, ten years old.
Do they miss their brothers and sisters, too? Bran wondered. Are they calling to Grey Wind and Ghost, to Nymeria and Lady's Shade? Do they want them to come home and be a pack together? (5.Bran.4)
Actually, it's not just Arya. The war between the Starks and Lannisters stems from both families wanting to reunite their families. Heck, even the Stark direwolves can't keep it together. The family structure is simply doomed in the Seven Kingdoms.
Her brother Viserys, Khal Drogo who was her sun-and-stars, even her unborn son, the gods had claimed them all. They will not have my dragons, Dany vowed. They will not. (13.Dany.17)
Family is so important to most characters that when they lose one, they often create a surrogate family to compensate. Dany names her dragons after members of her family, symbolizing the role they are filling. This surrogate family is very similar to Jon Snow's relation with the Night's Watch.
[Theon] shouted for a thrall to clean it up. Half my life I have waited to come home, and for what? Mocking and disregard? This was not the Pyke he remembered. Or did he remember? He had been so young when they took him away to hold hostage. (25.Theon.177)
Family is a source of comfort and survival, but it is also one of prestige and social class. When Theon is not welcome at Pyke, you'd think he'd just say, "Forget it then; I'm out of here." Yet for him to obtain his dreams, he requires his family's support. The vassals will not follow him as an individual but will only support him as a member of the Greyjoys.
Stannis pointed his shining sword at his brother. "I am not without mercey," thundered he who was notoriously without mercy. "Nor do I wish to sully Lightbringer with a brother's blood. For the sake of the mother who bore us both, I will give you the night to rethink your folly, Renly." (32.Catelyn.88)
Of course, the bros before not-bros rule isn't followed by everyone. Renly and Stannis both want to pursue their individualism by becoming king, and in order to do that, they must remove their family competition, a.k.a. each other.
"You take this business too hard, [Theon]. It is only that your lord father does not know you. With your brothers dead and you taken by the wolves, your sister was his solace. He learned to rely on her, and she has never failed him." (38.Theon.50)
Again, Theon's major paradox is that he cannot gain individuality without his family since social mobility in the Seven Kingdoms depends on the family right. Unfortunately for Theon, his sister, Asha, is the current family fav.
[Stannis] gave a shake of his head, like a dog shaking a rabbit to snap its neck. "Only Renly could vex me so with a piece of fruit. He brought his doom on himself with treason, but I did love him, Davos. I know that now. I swear, I will go to my grave thinking of my brother's peach." (43.Davos.92)
And my brother's peach is totally not a euphemism. Killing a family member is considered a huge taboo in Westeros society—again because of the importance the society places on family.
I cannot blame them, Catelyn thought. They do not know. And if they did, why should they care? They never knew my sons. Never watched Bran climb with their hearts in their throats, pride and terror so mingled they seemed as one, never heard him laugh, never smiled to see Rickon trying so fiercely to be like his older brothers. (56.Catelyn.3)
Many of the partiers have sworn loyalty to either the Starks or the Tullys, yet their loyalty is to the families. Catelyn seems to think that they do not know, nor do they really care, about the individuals making up that family.
"Would you ask a mother to sell one of her children?"
"Whyever not? They can always make more. Mothers sell their children every day."
"Not the Mother of Dragons."
"Not even for twenty ships?"
"Not for a hundred." (64.Daenerys.21-25)
Dany calling herself the "Mother of Dragons" and not the "Queen of Dragons" is also significant—it suggests that she views her power coming from her being the matriarch of her family. She cannot sell even one of her dragons because to do so would be to cut her power in third.
Outside, they made their farewells. Rickon sobbed and clung to Hodor's leg until Osha gave him a smack with the butt end of the spear. Then he followed her quick enough. Shaggydog stalked after them. The last Bran saw of them was the direwolf's tail as it vanished behind the broken tower. (70.Bran.95)
Okay, if you thought there would be a happy ending for any family in A Clash of Kings, guess again. No families manage to reunite and arguably the families are worse off than when the novel begins (possible exception: the Lannisters). While families provide protection and purpose, it looks as though several characters will have to find new forms of both to survive the novels to come.