Try to get them to update your first name in your profile at the Avis branch on your next rental. Similar process to how they'd update an expired DL or credit card on file.

Does the first name you want match your DL?

I'd tell Avis you have an updated DL and need them to change your information.

Also if the first name you are trying to use does not match your DL, it may be failing some validation check Avis does for your DL, and that may be why the change is not "sticking."