Welcome to FlyerTalk, jksim76.

AAA SoCal accepts debit, but not credit, cards online. Visa/MC Gift Cards are debit cards, so I can see how they might possibly work (although I haven't tried it yet myself). You can buy them with a credit card at your local Ralphs or wherever at the cost of $5.95 per gift card (which can be loaded with a maximum of $500).

But if AAA SoCal asks for the name on the card, how do you associate a gift card bought at a store with your name?

That shouldn't be an issue at a branch. However, if your insurance is over $500, or especially over multiples of $500, it could be odd trotting out gift card after card and paying $500 at a time!