Ned Stark loved all his children.
He wished the best for them all, but he had a special spot in his heart for his little wolf.
Arya was as wild as the day is bright. She forever be chasing after her older brothers and playing in the mud with her younger ones. With as much grief as this caused his lady wife for she always despaired that Arya was half wild and would never be a proper lady, never to find a lord husband, Ned could not entirely bring himself to stop it.
Her full bellied laugh was always sure to bring a smile to his face. Yes he knew he should do more to ready her for the life of a lady, to have her be more like her dear sweet sister Sansa just as Catelyn wanted. Then as soon as he would be on his way to scold her for skipping out on the septa's sewing lessons, he'd see her riding on a horse out pacing her brothers and suddenly he couldn't bring himself to say a thing.
He looked at Arya and he saw his own sister.
He saw Lyanna.
He saw her wild and free just as she always wanted to be.
He saw a dead girl gone too soon in his daughters place, and he worried.
She reminded him so much of his sister sometimes it hurt, she not only looked almost exactly like her, she also favored her in manor.
Two different girls, hopefully two different fates.
His worry is the reason he allowed her sword lessons once he had discovered her little needle, half in hope that her could do something for her which her was unable to do for his sister, and half out of fear that one day she may in fact need to know how to use a blade.
He wanted better for her.
He wanted to protect her from a cruel world.
In the end when he was lead to the stage as the people called for his head, he could only hope for her and her siblings, hope that he had done enough to prepare them.
Ned Stark loved all his children, but he was scared his little girl who wore a dead girls face.