Spoilers ahead for the most recent episode of Game of Thrones as well as A Song of Ice and Fire.
Ayra's going through a pretty brutal punishment — or, as the Faceless Men might call it, an education. After she went off-script and assassinated not the man she was assigned to, but someone on her personal revenge-oriented kill list, Jaqen H'ghar blinded Arya, leaving her to fend for herself on the streets of Braavos. Now, A Man is offering to give food, shelter, even her eyes back, if she would just say her own name. Could it be that simple? And why does she not accept the offer?
"Her training continues," explained Tom Wlaschiha, who plays H'ghar. Making Arya go blind wasn't just to punish her, but to force her into the next phase of training, "to sharpen all her other senses." The guild of assassins were not pleased with Arya's actions before, "because she really disobeyed them," but the larger concern wasn't so much that she didn't follow orders, but that she was still clinging to a remnant of Arya Stark. (Her revenge belonged to Arya Stark. She'll have to relinquish such attachments to continue on her path.) "Arya is still not 'no one,'" Wlaschiha said, "and that's the biggest obstacle."
During this phase of her training in the books, Arya is asked every morning, "Would you like to have your eyes back? Ask, and you shall see." Every morning she replies, "I may want them on the morrow. Not today." And every night at supper, she would be brought something to drink that renews the blindness for the next day, a potion drink with a queer, bitter taste: "Even the faint smell that warned her what it was before it touched her tongue soon made her feel like retching, but she drained the cup all the same." Even knowing it will make her blind, she drinks it, willingly, because to do so means she accepts that she has much to learn.
One lesson is to learn to hear. Already in episode one, just sitting there on the steps with her begging bowl, she could overhear fragments of conversation about the murder she had committed. "Are you listening, blind girl?" the Waif asks her. Sharpening her hearing makes her a better eavesdropper, but since Braavos is a port city, Arya requires a good dose of language lessons, as well — she must improve her Braavosi and High Valyrian, and the languages from Lys and Pentos, too. She continues to play the Game of Faces, also known as the lying game, but being blind means she has to learn to hear the lies, in tone and choice of words. With her hearing improved, her assessment of what's being said sharpened, and her knowledge of languages expanded, she becomes the perfect spy — completely ignored because she's a little blind beggar girl.
"Just blinding her will not lead her to give up," Wlaschiha said. "She'll continue to fight to get what she wants, but she's going to fight another way."
Another lesson Arya has to learn is to refine her sense of touch. She has to learn to recognize coins by their feel, to count steps, to balance in precarious places, the way Syrio Forel once taught her when she was chasing cats in the Red Keep. She also has to learn to uses her senses in combination — the way her footsteps bounce off the ceiling and echo around the hall, the way the pool of water affects sound. As she gets better at these things, her lessons start incorporating poisons and potions classes, even though this also means sometimes making herself sick. "Touch and taste could be perilous when grinding poisons," and sometimes even smell was not safe. She burns her fingers, blisters her lips, and has trouble keeping food down. "You have five senses," she's told. "Learn to use the other four, you will have fewer cuts and scrapes and scabs."
A girl won't have to be blind forever. But before she can accept the antidote, she needs to earn back the trust of the assassin guild, and show them that she is committed to becoming the perfect spy, the perfect actor, the perfect stealth killer. Perhaps then the Hall of Faces will be open to her once again. Even though some of their trade secrets involve a bit of magic, Wlaschiha said the focus is on craftsmanship and skill, and Arya is getting a crash course whether she realizes it or not. And someday, she may earn the right to wear another face again. "We have spare faces in our pockets, and we wait for the right moment to swap them. You should have four or five with you at all times!" he laughed. "You need to be prepared."