There are lot of exchange sites, and I think i saw one or two exchanging amazon gift cards too, but not sure
You can sell your gift card at eBay and thus the earned gift amount can be transferred into money and put into PayPal. But, you cannot obtain full 5$ from a 5$ amazon gift card. Because, if you list an amazon card of 5$ for selling it for 5$, no one will be interested in buying that. There will be plenty of offers lower to that (Some even lower than 3$). Additionally, there will be eBay charges to take care of.
here - found a site with a list of such gift-card-exchange links. Dont think they have any comparable offers to the card price... but anyway/.... http://www.giftcardadvocate.org/resources.html
I am willing to buy amazon gift cards using my paypal account. About US$200. If you have more I may consider. I do expect a discount. please email me if you have any.
WARNING!!!! There's buyer fraud on Ebay. Paypal and Ebay protects fraudulent buyers (who claims that they did not receive the codes or their account has unauthorized access). Sellers will very likely lose money. - Just ask any seller on Ebay. A certain level of experience is needed to differentiate bona fide buyers from fraudulent buyers. You must sell to buyers with track record of successful deals. You can sell to Rachael.zhang82. She has references of successful deals.