In a featurette for HBO, Tom Wlaschiha (Jaqen H’ghar) speaks about Arya’s second chance after he punished her for killing Meryn Trant in season 5 finale of Game of Thrones:
“For Jaqen, I think he does take a special interest in Arya. But to him, it doesn’t really matter who you were in real life. As soon as you enter the House of Black and White you’re under the system of the Faceless Men. So, it’s just another test to see how she will deal with this.
He blinded her in the first place, so it’s part of the game. He thinks he’s taught her a lesson and he wants to give her another chance. She has to go through all the tests to become the perfect Faceless Man.
He’s glad to see that the training’s working and that she overcomes certain things that were problems at the beginning of the training. She wouldn’t give up her name or her personality, and she didn’t quite understand what he wanted of her at first. So this is going better and better now.”
Check out the video below.