Even though they learned of the death of her aunt, surely there would still be someone there that could pay a small ransom. Both The Hound & Arya, and Sansa & Lord Baelish would recognize each other. At the very least, Arya would be safe, and The Hound would no longer need to feed or protect her. Later, The Hound told Brienne of Tarth that there is no safe place left for Arya, so why would he take her away from The Eyrie? They could have at least gotten food or rest, which would allow Arya to meet her sister Sansa, and likely cause her to stay for good. I'm referring to the TV show.
1 Answer
There are two reasons:
Nobody would recognize Arya if they went there. Arya never met someone from the Vale recently, so nobody could identify her. And it wasn't clear whether they would pay.
The bandits around the Vale have grown very strong. So during the journey it would be harder for him to protect her. SoS65
Furthermore, he doesn't go to the Riverlands and tries to sell her off there, because it was sieged at the time, so nobody could be distracted by a ransom.
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4@Chloe nobody knew it was Sansa. They all thought she was a niece or something of Petyr. As for Petyr, he wasn't the most trustworthy and also, he only met her once at the tournament of the Hand, so mistaking her is completely in the realm of the possible.– ArminCommented Apr 2, 2016 at 9:15
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3@Chloe but the hound doesn't know that this is Sansa. So this is another reason why he doesn't go there. The main reason still being the bandits.– ArminCommented Apr 2, 2016 at 9:27
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3Just saw this episode recently -- even in the show nobody knew Sansa was at the Eyrie, IIRC, so the Hound and Arya had no way of knowing Arya would be recognised (dramatic irony). Commented Apr 2, 2016 at 9:37
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2@Namfuak The only reason the Hound and Arya could have known Petyr was at the Eyrie would be if they had somehow heard about Petyr's recent nuptials. If I'm remembering events correctly, I believe even the Lannister network of eyes-and-ears was slow catching rumors of Petyr's whereabouts, so I doubt Hound & Pup managed to stumble upon this kind of info while bumbling along the countryside. :)– David HCommented Apr 2, 2016 at 14:57
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2@DavidH That's what I figured, so all the discussion of whether they thought Petry would pay is moot, since they didn't think he was there in the first place. Commented Apr 2, 2016 at 17:13