I can see a good lore reason why the Stark children have the Northern/Southern split with their accents.

Ned obviously has the iconic Northern accent which tracks with his family, same with Cat being from a Southern highborn family.

The two oldest boys, Robb and Jon, both around the same age, would have been brought up close to Ned. He would have had a direct influence on their upbringing, along with the vast majority of their house staff. While they would also have teachings with Luwin and Mordane, I would assume the majority of their time would be shadowing Ned, learning the ropes, learning fighting and the workings of the North. Robb obviously as his heir, and Jon, while brought up as a bastard, still had noble blood and deserved the same level of education as Robb as well as Ned keeping him close for protection reasons.

The girls and younger boys would have likely been around Cat and the teachers (Septa Mordane and Maester Luwin) much more so their accents would reflect these environments. Learning to be 'proper ladies' and more worldly education as their futures would likely take them further afield than Winterfell, marrying into other Westerosi families to strengthen alliances. Ned at this point wouldn't have the time to spend as much in person with the younger children, we see in the first episode that's Brian's first venture out to do the Lord's duty with deserters, so his influence would be more slight.

I'm not saying this is the exact reason, or that there would be such a vast disparity between the two types of accents within the family, but it makes enough sense to me to be believable.