I'm currently watching Matt Staples on twitch and he put an opponent in an interesting situation which led me to this hypothetical - but first lets talk about Matt's hand.

Matt had JsJc in the hole. I don't remember his position, but he was OOP to villain. I believe there was a 3B prior to the flop.

Flop was 9h3hJd. Matt check raised and was called.
Turn was Kd. Matt bet 33% of pot (I think) and was called.
River was a brick - maybe 4c. Matt decided to try to induce a bluff with a check, but villain checked behind. Ultimately villain had TT.

But that got me thinking - what do you do on the river if you're villain and you're holding something like AQhh? Do you bluff? If so, how much? Or do you check behind and essentially concede the hand unless the other guy is holding air?

I think you should mix strategies so you don't become too predictable. If you have a read on the other player and you know they like to trap in that situation, check behind. If you don't know much about the other player, sometimes bluff and sometimes check. Some players even know the percentage of times they should do each. I just go with my gut every time depending on how the hand was played to the river.

The guy above nailed it. A good mix of both is the right approach. It really on depends on your opponent, and the action in the hand. If you know your opponent is a calling station, or will trap, do not bluff! If the opponent is known to check rivers with marginal hands or missed draws, then a good sized bet can occasionally take it down. Bet sizing is VERY important in these spots as well.

I would have checked as well. Simply because in order to pull off a bluff you need to "tell a story" of having a great hand. The villain was weak and played weak so there is no way a bluff would work, even if the hero only had mid or top pair.

I think the check raise on the flop would deter me from trying to bluff in that spot . Even though there are some strong possible hands with that board , the check raise on the flop means he is strong and would probably call any bet . I would concede the hand and move on .