So much of Arya’s story is about challenging the constructed and intrinsically linked ideas of gender and beauty, though maybe more importantly it is also about exposing the loss of self and individuality that happens both when you do conform to these standards but also when you try to reject them, i.e. when society doesn’t have a place for you at all. And rather than accept either of these fates that are presented to her, Arya is being set up to instead learn to carve out a space of her own.
I think this arc is foreshadowed all the way back when she first is learning water dancing with Syrio:
"Boy, girl," Syrio Forel said. "You are a sword, that is all."
Throughout Arya’s journey, it becomes clear that she doesn’t really fit either the description of being a boy or a girl, and being unable to find acceptance for who she truly is, she then instead ends up with The Faceless Men, where she will be stripped of her identity entirely, becoming “no one”, simply an instrument (a sword) in the hands of others.
But even when Syrio first puts forth this notion, Arya also refutes him or at least makes her own addition to the idea:
He clicked his teeth together. "Just so, that is the grip. You are not holding a battle-axe, you are holding a—"
"—needle," Arya finished for him, fiercely.
She isn’t just any nameless sword, she is Needle.
And when she reaches The Faceless Men and is asked to give up every one of the last few possessions that remain to her, Needle is the one thing she can’t let go of:
She stood on the end of the dock, pale and goosefleshed and shivering in the fog. In her hand, Needle seemed to whisper to her. Stick them with the pointy end, it said, and, don't tell Sansa! Mikken's mark was on the blade. It's just a sword. If she needed a sword, there were a hundred under the temple. Needle was too small to be a proper sword, it was hardly more than a toy. She'd been a stupid little girl when Jon had it made for her. "It's just a sword," she said, aloud this time . . .
Needle was Robb and Bran and Rickon, her mother and her father, even Sansa. Needle was Winterfell's grey walls, and the laughter of its people. Needle was the summer snows, Old Nan's stories, the heart tree with its red leaves and scary face, the warm earthy smell of the glass gardens, the sound of the north wind rattling the shutters of her room. Needle was Jon Snow's smile. He used to mess my hair and call me "little sister," she remembered, and suddenly there were tears in her eyes.
It represents the true core of her identity, her history, her family and allegiances, her home, her sense of justice. And though she is inching closer every day towards becoming “no one”, Needle still remains hidden away in a crack in the wall, waiting for her to come back and claim it.