Unlike most others in this thread, apparently, I think the way the rental companies handle this is preposterous and abominable.

And I say that as a 15-year industry veteran.

Also, as a 15-year industry veteran, I think I can fairly assess things and say that if you actually presented your situation (or any other number of situations, like a delayed/canceled flight, etc.) to any high-ranking executive in the rental industry, they would agree that it's preposterous that you should be charged more through no fault of your own when you're consuming less product and shouldn't be that way. Unfortunately, the upper muckety-mucks don't know what's going on (in the practical outworking of the high-level ideas/policies they set) and the middle-to-low people lack the intelligence to sensically execute said policies. (The rental car industry doesn't attract the best and the brightest.)

You mentioned that you're renting from a small airport. I would pursue two avenues:

1) Ignore the clueless and powerless outsourced agents at Avis's reservations line. Reach out to the location directly--preferably a supervisor or manager and let them know your flight was canceled and you'll be coming in late and please can they hold the reservation for you? That has a pretty good chance of working.
2) If that doesn't, push your case up to a higher-level executive. You can find some contacts here.

Voluntary changes are one thing, especially where they're designed to circumvent yield management restrictions, but involuntary changes like flight cancellations ought to be handled much better by the rental companies. Someone I worked with recently had a canceled flight with Sixt and ended up having to pay $350 more because she called them and said she couldn't pick up until the next morning. Shame on Sixt.