Cat is a funny character, because her impact is felt more with how she raised her kids, then the decisions she made during the show. Ned inspired his kids. He was clearly a more hands on father then most in Westeros, especially if you look at Jon and Arya. They take after him something fierce. Arya is basically Lyanna, Ned's sister reincarnated. Robb tries to be like his father.
The Cat that raised her kids, is not the same person we see on the show. She was far more Tully then she is now, though you can still see the selfishness, the pettiness, the dramatics at times. Living in the North all those years changed he, but you can still see it.
Robb could have used Jon at his side during the war. That would have never happened because of Cat. The reason he had to go to the Wall was Cat. He was not allowed at King's Landing, and Cat forbid him from staying at Winterfell with Ned gone.
One of the reasons I prefer the show relationship between Robb and Talisa, is because of how it reflects on Cat. In the book the honor bit is reversed. Robb marries the book character out of "honor". On the show it is the romantic ideal of love. He can't live without her, so he breaks his vow, betrayals his code. He ruins his honor, and loses the war, for what he wants. Cat instilled this in her kids and you see it in Robb, Sansa and Bran.Thinking about what you want rather then doing "what is right". It is why she releases Jaime. So Cat is left with what she has molded Robb into, and because he is king, like Cersei, she is powerless to make him do anything.
In the end Cat loves her kids more then anything, but she is selfish, and it rubbed off. That is why she gets a hard time for how she treats Jon. It reflects her personality. Her willingness to throw a child aside for what she wants. It isn't like Jon is unlikable. He is everyone's favorite sibling outside of Sansa, and why is that? Because she took right after her mother.