It's been a long time since I read the first few books, but I was re-reading A Clash of Kings just before reading A Dance with Dragons, and I noticed that "The Hound" (Sandor Clegane) is no longer part of Arya's prayer in the new book.
What changed to make her "forgive" him?
The list is of people she wants revenge against. In A Storm of Swords, when Sandor Clegane was greviously wounded, she denied him the gift of mercy and (apparently) left him for dead. That's when she dropped his name from the list.
When Arya and Sandor were wandering Westeros aimlessly after the Red Wedding she used to say her prayer with the Hound in it regularly. After a while, a sense of camaraderie developed between the two (but not friendship) and Arya started to notice that she had started unconsciously to leave out the Hound's name out of her prayers, and she never bothered to re-include him.
From the show's perspective Ayra admits that she did remove The Hound from her list. She was confused about her feeling towards Sandor Clegane after the adventures. While we may not know the exact moment this happened, the whole story lines allows Ayra to realize that not all people can be defined as simply "good or evil".
Jaqen knew Arya was lying when she said she hated The Hound.
(Starting at about 0:12, emphasis mine)
Ayra: A man called The Hound, Sandor Clegane. He was trying to sell me, but was wounded in a fight. He begged me to kill him, but I wouldn't. I left him in the mountains to die. I wanted him to suffer, I hated him - - -
Jaqen hits Ayra with the stick to indicate she is lying
Ayra: (with stronger inflection) I hated him!
WHAP
Ayra: (raising her voice) That's not a lie!
WHAP
Jaqen: A Girl lies. To me, to the Many-Faced God, to herself. Does she truly want to be No-One?
Ayra later admits she took him off her list.
(Starting at 1:09 in the linked video, emphasis mine.)
Waif: Tell me about The Hound.
Ayra: Also dead. Ayra Stark left him to die. He was on her list.
Waif whacks Ayra with the stick to indicate she is lying
Ayra: He was not on her list anymore. She'd taken him off it.
Waif: Why? Didn't she want him dead any longer?
Ayra: She did and she did not.
Waif: She sounds confused.
Ayra: Yes, she was.
In addition to the above, if you will note on that ARYA/SANDOR road trip, she always calls him the Hound - until a certain point. and then she starts calling him Sandor. She is also realizing that the bad things that she attributed to him were actually required of him in his post as Kingsguard. He has broken that by escaping. She is realizing that there is more to this man, some good. He is becoming more human to her, and not a malicious burned monster. Sandor has a huge role in this series.
