0 Confirmation transactions are totally spendable without any problems. Your client might now allow you to do this but that is a problem of your client, not the bitcoin protocol. So if you have a non-confirmed transaction you could spend the output and add a fee to increase miners incentive to mine both and thus get the fee.
You can try this yourself to confirm that it doesn't work. Send a transaction to an address that you control with zero fee (but a transaction that is allowed to have zero fee) and then wait till you see that it was picked up by the bulk of the network (check blockchain.info). Next, rebroadcast a fraudulent transaction using the same coin to a different address and this time attach a higher fee (you can do all of this with the "transactions" page at brainwallet.org). The second transaction won't propagate, and the miners won't mine it, because they can see that that output has already been spent.
As far as I know, the only ways you may get the second transaction to confirm would be to (1) be in cahoots with a miner who controls a lot of hash power and have him manually add your fraudulent transactions to his memory pool, (2) hope that your first transaction gets dropped by the network before it gets mined (and that no-one rebroadcasts it). After your first transaction has been dropped by enough miners and nodes, your second transaction can propagate.
I think you misunderstood. What he wanted to say is you can spend unconfirmed coins (legitimately) and include a fee in the transaction so the previous transaction that would take long to confirm would confirm quicker - because it's in miners best interest to include both.
Oh, yes you're absolutely right, I misunderstood. I'm used to people arguing that the double-spend problem is bigger than it really is, and I thought this was one of those cases. My apologies, Aswan.
