They occupy similar positions in their respective families. Most consequentially, they both have weird closeted Uncle Dad relationships with their sisters’ kids. They’re also middle children. Much as Jaime gets legal precedence as Tywin’s first son, he is Cersei’s little brother.
More abstractly, I think Ned and Jaime are at contrasting ends of the same wavelength when it comes to inhabiting personas. It’s not that either of their reputations are unearned: Ned is more or less a stand-up guy, and Jaime really is a treacherous piece of work who’s willing to undermine the king by breaking every one of his vows. But those reputations are more complex than they appear; at best, they’re right for the wrong reasons. Ned isn’t a good guy because of his publicly rigorous rule-following, but because he is capable of putting other people’s well-being before his own image. Jaime killing Aerys wasn’t a coup, though his affair with Cersei very much is. And they’re both really practical about using those reputations, particularly in the interests of conflict avoidance.
“Aleena? No. You told me once. Was it Merryl? You know the one I mean, your bastard’s mother?“
“Her name was Wylla,” Ned replied with cool courtesy, “and I would sooner not speak of her.”
“Wylla. Yes.” The king grinned. “She must have been a rare wench if she could make Lord Eddard Stark forget his honor, even for an hour. You never told me what she looked like …”
Ned’s mouth tightened in anger. “Nor will I. Leave it be, Robert, for the love you say you bear me. I dishonored myself and I dishonored Catelyn, in the sight of gods and men.”
“Gods have mercy, you scarcely knew Catelyn.”
“I had taken her to wife. She was carrying my child.“
Must you make me say the words? Pia was standing by the flap of the tent with her arms full of clothes. His squires were listening as well, and the singer. Let them hear, Jaime thought. Let the world hear. It makes no matter. He forced himself to smile, “You’ve seen our numbers, Edmure. You’ve seen the ladders, the towers, the trebuchets, the rams. If I speak the command, my coz will bridge your moat and break your gate. Hundreds will die, most of them your own. Your former bannermen will make up the first wave of attackers, so you’ll start your day by killing the fathers and brothers of men who died for you at the Twins. The second wave will be Freys, I have no lack of those. My westermen will follow when your archers are short of arrows and your knights so weary they can hardly lift their blades. When the castle falls, all those inside will be put to the sword. Your herds will be butchered, your godswood will be felled, your keeps and towers will burn. I’ll pull your walls down, and divert the Tumblestone over the ruins. By the time I’m done no man will ever know that a castle once stood here.” Jaime got to his feet. “Your wife may whelp before that. You’ll want your child, I expect. I’ll send him to you when he’s born. With a trebuchet.”
Silence followed his speech. Edmure sat in his bath. Pia clutched the clothing to her breasts. The singer tightened a string on his harp. Little Lew hollowed out a loaf of stale bread to make a trencher, pretending that he had not heard. With a trebuchet, Jaime thought. If his aunt had been there, would she still say Tyrion was Tywin’s son?
The two characters are retreating to different personas in these passages. Ned performs as Mr. Stuffy No-Sex Honor-Man while Robert of all people keeps pressing; Jaime’s all GRRR I AM THE KINGSLAYER QUAIL BEFORE MEEEEEE. (He’s so proud of that trebuchet flourish.) But they’re both leaning into what other people think of them, using dishonorable methods to honor a promise made to a dead woman, a promise that, if known, would undercut the reputations they use to carry it out.
Those similarities make them a pretty solid nature/nurture comparison. Ned was born in Winterfell, to a family that at least seems to have been somewhat functional, and then given to Jon Arryn in his boyhood. Jaime was born to one psychopathic supervillain, and then drafted by another in his boyhood. Would Jaime have been what he is if he’d had half of what Ned did in that regard? Look at how – in a few short months, as a fully-baked adult, with a lifetime of sunk costs into his many transgressions – he takes to the combination of Brienne’s good example and some professional and psychological autonomy once Tyrion does the world the favor of punching Tywin’s ticket. That’s not entirely down to Brienne being some kind of moral moral muse who ~makes him wanna be a better man or even to Tywin’s iron grip on his children’s world. It’s about Jaime being not only responsive to outside influence, but being disproportionately receptive to good influences after a lifetime of being dominated by bad ones.
And all of that kind of winds around to that pivotal moment in the throne room at the end of the Rebellion.
He was seated on the Iron Throne, high above his knights……I was still mounted. I rode the length of the hall in silence, between the long rows of dragon skulls. It felt as though they were watching me, somehow. I stopped in front of the throne, looking up at him. His golden sword was across his legs, its edge red with a king’s blood. My men were filling the room behind me. Lannister’s men drew back. I never said a word. I looked at him seated there on the throne, and I waited. At last Jaime laughed and got up. He took off his helm, and he said to me, ‘Have no fear, Stark. I was only keeping it warm for our friend Robert. It’s not a very comfortable seat, I’m afraid.’
Ned’s belief that he “forced” Jaime off of the Iron Throne with his Power of Heart™ is rather charming to me. As if Jaime stopped mid-coup and had his men stand down while Ned made his dramatic entrance. Less charming, of course, is what stuck with Jaime:
Bolton’s silence was a hundred times more threatening than Vargo Hoat’s slobbering malevolence. Pale as morning mist, his eyes concealed more than they told. Jaime misliked those eyes. They reminded him of the day at King’s Landing when Ned Stark had found him seated on the Iron Throne.
This is kind of a startling comparison the first read through, but it really pops the second time around. At this point in the narrative, Roose is actively plotting against Ned’s son. Like Jaime, Roose will kill his king. Unlike Jaime, Roose will violate an oath he swore as an adult and of his own free will, and will do so strictly for his own benefit. Jaime’s comparison of Roose to Ned is deeply unsettling.
More than anything else in Jaime’s POV recollections, it’s this comparison that clarifies why Jaime chose the course he did at that moment and didn’t tell anyone why he’d killed Aerys. In terms of keeping his head on his shoulders, it was probably his best possible option. Trying to justify himself would have been submitting to Ned’s judgment. Trying to justify himself without proof, which he did not have at the moment, could easily have backfired. Ned, right then, was in sore need of someone to blame for all of the horrible things he’s experienced and seen. I wouldn’t trust him with the truth either, this stranger with Roose Bolton’s eyes. Jaime, disillusioned and damaged as he was, was at a critical juncture where he could have pulled out of the dysfunctional Lannister spiral. But when an unstoppable force meets an immovable object, it bounces back just as hard, in the opposite direction.