I think we don't talk enough about how Jon Snow secretly had a sword made for Arya at Winterfell - without anyone knowing! And that this was something he was planning on for a while, with the intention to teach Arya some fundamental sword skills - without anyone knowing!!
It reminds me about how much Arya must have poured out her heart and soul to Jon Snow about EVERYTHING, considering how much Jon knows about her. The very best of confidantes who guarded their secrets with each other and are the most loyal of siblings.
It was to Jon Snow that Arya goes, after being bullied for her looks, worried that she too was a bastard and Jon who consoled her (ignoring his own pain at being one). It's Jon who praises her as pretty and clever and understands that deep curiosity and ambition in her.
It's Jon who understands that Arya is interested in something different and that this is also deserving of attention. The ONLY person in the whole of Winterfell - not her parents, her other siblings, her teacher. Only Jon Snow.
I can imagine Jon and Arya just hanging out in a quiet corner of the Godswood, under the weirwood, with Arya pouring out her frustrations and chatting about playing with the serving girls and Jon talking about his day practicing the sword. They know each other so well, that they are famous for finishing each other's thoughts. They share such a singular bond that he even got her sword name right!!
Arya seemed puzzled at first. Then it came to her. She was that quick. They said it together:
"Needle!"
The memory of her laughter warmed him on the long ride north. - Jon, AGoT
Making Needle wouldn't have been easy considering it had to be done secretly. Clearly Jon thought that both his father and Catelyn wouldn't have been happy if they knew that the bastard was having swords made for their daughter.
"Give it to me."
Reluctantly Arya surrendered her sword, wondering if she would ever hold it again. Her father turned it in the light, examining both sides of the blade. He tested the point with his thumb. "A bravo's blade," he said. "Yet it seems to me that I know this maker's mark. This is Mikken's work."
Lord Eddard Stark sighed. "My nine-year-old daughter is being armed from my own forge, and I know nothing of it. The Hand of the King is expected to rule the Seven Kingdoms, yet it seems I cannot even rule my own household. How is it that you come to own a sword, Arya? Where did you get this?" - Arya, AGoT
Jon Snow took the time to research swords that Arya could hold and handle. He must have been up in Maester Luwin's turret looking through books for the design and asked questions of the Winterfell master-at-arms Rodrik Cassel about Braavosi swords.
She giggled at him. "It's so skinny."
"So are you," Jon told her. "I had Mikken make this special. The bravos use swords like this in Pentos and Myr and the other Free Cities. It won't hack a man's head off, but it can poke him full of holes if you're fast enough." - Jon, AGoT
He'd had Mikken make a sword for Arya once, a bravo's blade, made small to fit her hand. Needle. He wondered if she still had it. Stick them with the pointy end, he'd told her, but if she tried to stick the Bastard, it could mean her life. - Jon, ADwD
It had been so long since he had last seen Arya. What would she look like now? Would he even know her? Arya Underfoot. Her face was always dirty. Would she still have that little sword he'd had Mikken forge for her? Stick them with the pointy end, he'd told her. Wisdom for her wedding night if half of what he heard of Ramsay Snow was true. Bring her home, Mance. I saved your son from Melisandre, and now I am about to save four thousand of your free folk. You owe me this one little girl. - Jon, ADwD
After getting the idea of what kind of sword works for Arya's small hands, Jon then goes to Mikken, requesting that he make a small Bravo's blade. I feel certain that Mikken had no idea that he was secretly having a sword made for the Lord of Winterfell's daughter. I wonder what Mikken's thoughts were on Jon Snow wanting that specific blade made. He clearly did not think it important to mention to Ned. And no one knew - not Robb or Theon or even the Winterfell master-at-arms!
Given how sudden the whole deal was with Ned leaving for King's Landing, IMO, it's clear that Jon was planning on secret rendezvous with Arya where he could show her the basics of using a sword. Jon is certainly no Syrio Forel and Arya certainly learned more from an actual Bravo master fencer than from Jon Snow.
And yet just knowing that Jon had Needle secretly made and was planning on secret lessons for Arya because he knew just how desperate she was to learn something different, something unacceptable for Winterfell's daughter and that he did so at the great risk of displeasing a father he looked up to and the Lady Catelyn Stark who already wanted him gone.
He truly is Lyanna's son in every way that mattered.