This won't help you for this one, but I also always use delivery confirmation for my amazon or ebay sales. If it's an expensive item I also always include insurance. I just figure it as part of the cost I set and never ask the buyer to make a decision. I started doing this after one of my first sales disappeared. I had the delivery confirmation slip to prove it in that case, but I think I was being scammed because it was something they could resell for a lot more than they paid me.
Double check the address that you sent it to. I always cut and paste whatever Paypal says.
Suggest that your buyer ask any other family members or neighbors, who sometimes set things aside without informing the recipient. I know I stashed something once and reported it missing during a busy holiday season. I've also had neighbors intercept my package. Once my mom addressed something to me at my old address--one block over.
If the buyer doesn't come up with anything on their end after a trip to the post office, I think you ought to offer a full refund. One negative feedback will hurt you a lot on ebay, unless you're a big seller. You can make an inquiry at your post office, but without a tracking number they won't have much to go on.