All of these other sitters are right on the money!
A little flexibility is expected when travel plans change, but you certainly don't make money by letting clients write the rules of your home business.
If the flight causes the owner to return on the next day, a full overnight cost should be paid. If it will be several hours later than originally anticipated, I would offer a daycare rate to the client, or a prorated daycare rate to cover the additional care. Typically, in a similar situation, I send a message like:
"Hey (Client's Name)! I'm sorry to hear that you're having trouble with your flight. I'm happy to continue to watch (Pet's Name) until you're able to make it back. Because the delay will cause you to arrive outside of my pick-up hours, I can send over a prorated daycare booking to you. This will ensure that (Pet's Name) continues to be covered by insurance until your return. My dog and (Client's Pet) are having a ton of fun together, so I know they won't mind a little extra playtime. If you're not comfortable with the extra charge, please let me know, and I can make myself available to have a friend or family member pick him up from me instead. I'm happy to help in whatever way you're most comfortable!"