Jon Snow and Arya Stark are the only ones of the current generation to have the Stark look - the long face, brown hair and grey eyes. They are also the two Starks who think of and mention Ned Stark the most, follow his guidance and advice. They often mention their ‘Father’s bannermen’ and actively participate in plots to help house Stark despite not having a lot of power or agency to do so (Arya at Harrenhal and Jon Snow with Stannis and the Karstarks). They are the two Starks who embody Ned’s vision of house Stark the most.
Jon remembered something his father had said once. A wall is only as strong as the men who stand behind it.
A lord may love the men that he commands, he could hear his lord father saying, but he cannot be a friend to them. One day he may need to sit in judgment on them, or send them forth to die.
“The map is not the land, my father often said.
“My lord father said he never ate half so well as when visiting the clans.”
“My lord father used to tell me that a man must know his enemies. We understand little of the wights and less about the Others. We need to learn.”
“My lord father used to say a man should never draw his sword unless he means to use it.”
He might have known them anyway, just by the way they stood. A good lord must know his men, his father had once told him and Robb, back at Winterfell. - Jon, ASoIaF
I should kill them myself. Whenever her father had condemned a man to death, he did the deed himself with Ice, his greatsword. “If you would take a man’s life, you owe it to him to look him in the face and hear his last words,” she’d heard him tell Robb and Jon once.
Tell me what to do, you gods,” she prayed.
Then, so faintly, it seemed as if she heard her father’s voice. “When the snows fall and the white winds blow, the lone wolf dies, but the pack survives,” he said.
Last night she’d had a bad dream, a terrible dream. She couldn’t remember what she’d dreamed of now, but the feeling had lingered all day. If anything, it had only gotten stronger. Fear cuts deeper than swords. She had to be strong now, the way her father told her.
Her father used to say that a lord needed to eat with his men, if he hoped to keep them. “Know the men who follow you,” she heard him tell Robb once, “and let them know you. Don’t ask your men to die for a stranger.” - Arya, ASoIaF
It’s Jon and Arya who idealize Ned Stark the man, who measure their actions against their father’s and often times come up short.
“Jon did not know that he could tell honor from shame anymore, or right from wrong. Father forgive me.” - Jon, ASoS
Arya told of Yoren and their escape from King’s Landing as well, and much that had happened since, but she left out the stableboy she’d stabbed with Needle, and the guard whose throat she’d cut to get out of Harrenhal. Telling Harwin would be almost like telling her father, and there were some things that she could not bear having her father know. Arya, ASoS
It’s Ned who gives Arya courage and strength when she falters, who tells her “When the snows fall and the white winds blow, the lone wolf dies but the pack survives.” Words she follows as she tries to find her pack. It’s Ned’s teaching and advice that Jon uses to plan Stannis’ campaign in the North.
Arya’s relationship with her mother is more complicated by Catelyn’s desire for her to be a fundamentally different person. Jon does not have a mother figure in the story, however, it’s interesting that growing up the only positive female influence in his life is Arya Stark (Well, apart from Old Nan - though I doubt Old Nan did much parenting apart from storytime!).
This is subconsciously reflected in the girls he admires, befriends and loves (He compares Ygritte physically to Arya, he compares Ygritte and Alys’ personality to Arya’s) as much as his contempt for Catelyn is reflected in the disdain he shows for what he considers to be the typical Westerosi lady who hold high regard for propriety and societal norms.
A warrior princess, he decided, not some willowy creature who sits up in a tower, brushing her hair and waiting for some knight to rescue her. - Jon, ADwD
GRRM seems to be playing around with an Oedipal Complex for Jon Snow in that he keeps comparing the girls he is attracted towards and admires with Arya and Arya is a mini Lyanna according to the characters who know both Arya and Lyanna. Jon and Arya are close because Arya is like Lyanna in terms of personality and looks. Jon of course does not know this, however, he is more or less getting to know his mother through his relationship with Arya. This is the GRRM approved world book art for Lyanna and Arya:
Arya Stark in KL Lyanna Stark at Harrenhal
“Lyanna might have carried a sword, if my lord father had allowed it. You remind me of her sometimes. You even look like her.” - Arya, AGoT
It’s Lyanna’s son who admires Arya for who she is, Lyanna’s son who defies his father, secretly arms Arya and gifts her a sword - Needle, symbolic of the bond between them.
As an aside, Arya (A Lyanna look alike) has the closest bond with Rhaegar’s son, Jon Snow and also has a friendship and something brewing with Robert’s son Gendry Waters while on the run, a point to note when one considers what happened with Lyanna, Robert and Rhaegar.
Mothers and sons, fathers and daughters. Something to consider about these bonds of love and friendship between the younger generation when one looks back on the everything that happened in the past.
