Minus Robb, all the kids inherited Catelyn’s personality
I think all the kids are a mix of their parents.
Robb, TBH, I find indistinguishable between his parents. His education is clearly guided by Ned, but he began the books as a laughing, somewhat daredevil of a youth. Reminds me more of either the Tullys or Ned’s older brother Brandon, TBH. He has a strong sense of duty, which is like both his parents.
Sansa has Catelyn’s propriety and sense of decorum and presentation, but her idealism is Ned to a tee.
Arya is driven by a strong if childlike sense of justice, reminding me of Ned being so angry that he couldn’t solve the Arryn murder mystery and put things aright. But her temperament is stubborn, angry and proud. She reacts to loss much more like Catelyn does, and she has the same irritation with other people who fail to help her personal agenda as her mother.
Bran is known to be affable and generous, actually reminds me a lot of Cat’s brother Edmure, but he’s also more quiet and contemplative, a little like Ned. He’s more curious about the outside world than how I think of Ned, but he has that dreamer’s quality and propensity to romanticize knights the same way Ned talks about Arthur Dayne, tbh.
Rickon, well honestly who knows about Rickon. He wants his family. That’s pretty much both Stark and Tully.
Sansa and Bran have a lot of Ned’s quietness and dreaminess IMO, while Robb and Arya are more expressive and concrete. Robb is being taught by Ned to follow in Ned’s footsteps but he is actually a first-born like his mother. Arya aligns with the Stark ethos but her temperament is more like her mother’s than her father’s. Sansa and Bran were the children who were attracted most to their mother’s faith, but we see over the course of the novels that the old gods mean something to them too.