NFC Bitcoin payments are a thing I have been dreaming of since before Bitcoins or NFC were invented.
The ideal solution here is to keep a private key on the ring, and actually have the ring itself perform the ECDSA signing, and then it's up to the merchant to propagate the transaction on the network. The one (HUGE) problem with that, is without any sort of display, or verification mechanism, the ring is just blindly signing any transaction requests sent to it, so anyone could just walk by and quickly grab all the coins out of the ring.
This version of the ring would not be ideal for these types of bitcoin payment. You still need at least a display to show how many bitcoins are going to be transferred, and physical buttons to approve/reject the transaction, and ideally a way to enter some sort of PIN. I imagine more of an NFC necklace for these complex transactions.
That said, there are plenty of awesome bitcoin related things that can be done with this version of the ring.
Team up with Andreas Schildbach (creator of the most popular android wallet) and have him build in an NFC locking mechanism. Keys would be encrypted in the wallet, and would be unlocked via a simple locking mechanism (similar to phone unlock). This is not very fancy cryptographically, and there are plenty of weak points to attack, but it would work, and it would be a good additional locking mechanism.
The real magic would be if Andreas could figure out how to keep a bitcoin private key on the ring, and have special NFC addresses in the wallet app. If you are sending from an NFC wallet, you have to tap your ring to send the transaction, and the transaction is signed on the ring itself. It's very important that the key is actually locked with a random PIN only accessible by the bitcoin wallet app, or anyone else can come along, like I said before, and grab the coins that way. If done right, the private key would be generated on the ring, and it would never need to leave the ring.
Team up with various web wallets and exchanges to implement an NFC authentication system. You load in your public key to the site, and they send down a signing request that you have to sign before you can do certain actions. This can be done already with client SSL auth, but I think some sort of json API would be more developer friendly. This suggestion really has nothing to do with bitcoins, but there happens to be a lot of bitcoin wallets and exchanges, as well as users who are demanding stronger authentication technologies.