Yes. At many Avis locations, they don't actually process your rental fully until exit the gate. So it will be a Saturday morning pickup in that case, even if you pick the keys on Friday night. And so your rate may be changed because you "picked it up" so late.

I suggest instead you find the closest hotel to the airport with free parking. You haven't mentioned which airport this is, so we can't give you recommendations for one. But at many airports, just a little further away and the hotels start to have free parking. I've never had an issue finding a close-by airport hotel with free parking any time I wanted to pick up the car before going to the airport hotel.