The patriarchs of both the Corleone and Stark families make some very critical mistakes which bring both their families to the brink of utter and complete destruction. Neither of them realizes that the world has changed in some very fundamental ways since they took power. Therefore each of them is a little too secure in his power and doesn't realize that other people are playing by a different set of rules.

Ned Stark doesn't recognize that Queen Cersei or the relatively low-born Littlefinger can actually be threats. So he, from misguided mercy and benevolent sexism, informs Cersei that he intends to act against her, never imagining that she will dare to harm him or the king, in order to keep her secret safe. He makes his disgust with Littlefinger clear and refuses to take Littlefinger's advice. Ned even states that he will act in a manner which is clearly opposed to Littlefinger's best interests. He is nonetheless shocked that both Littlefinger and Cersei act in ways which, while dishonorable and underhanded, advance their own interests to Ned's fatal cost.

Vito Corleone doesn't suffer from mercy or honor but he does have Ned's pride. And it's this pride which makes him think he can impose his personal distaste for drugs onto the entire NY Mafia. It's pride which makes him overestimate Luca's abilities and his own craftiness, thus throwing away his most valuable warrior even before the war begins. It's pride (an unwillingness to look into details?) which makes him not realize that Paulie is a traitor until it's too late. He doesn't pay the same price that Ned does but the impact is pretty much the same. Like Ned, his oldest son has to take over a few years before he's ready.

Robb Stark is like his father in just about every way. He might even be a little more dangerous on the battlefield. He never loses a battle. Like Sonny is described in the book, Robb Stark is a relentless executioner who doesn't mind getting his hands bloody. But just like Santino Corleone, he lacks the ability to see the big picture. Finally his enemies use his strength (his honor and need to play by the rules) against him. Trying to make amends for a broken betrothal Robb is betrayed in a very ugly way. He never saw it coming.

Similarly Sonny possesses all of Vito's strength and then some. Although calling Sonny honorable might be a stretch he does believe in protecting the younger and weaker members of his family, a positive quality that much like Robb's honor is calculatedly turned against him and used to bring about his demise.

In The Godfather the Corleones had Michael to step in after what looked like total disaster, get revenge and make the Family even stronger than before. He had the benefit of learning from his father's and brother's mistakes as well as getting a crash course in being a boss from his Dad.

In The Game of Thrones series, it's unclear as to whether Martin wants to even go down the Stark revenge path. He enjoys subverting expectations, maybe too much so at times. The remaining Starks are very young but each will have powers which I won't discuss here just yet. Martin has not finished the series so no one except Martin and the HBO show runners know what the ultimate plans are. It may well be though that the Starks will get some measure of justice/revenge/rebirth. Their totem animal is the direwolf. The initial title for the final book in the series was "A Time for Wolves".