On the Avis website, go to Customer Service -> FAQ -> USA -> Reservations -> Can I pay with a debit card (sorry, you can't paste a link directly to the page you need). There's some useful info, including:

"Avis reserves the right, in its sole discretion, to seek a Debit Card authorization hold in excess of the estimated rental charges. When using a debit card at Avis, there may be a minimum hold of $500 and a maximum hold of the estimated rental charges will be placed on your account.
Upon returning the vehicle, Avis will process a release of the unused portion of the hold subject to your bank's procedures. The hold may take up to 2 weeks to be released by your bank."