I would start by filing a 'lost mail claim' on behalf of your customer. If you had insurance or shipped Priority Mail then you can file an insurance claim. If you printed the shipping label through Etsy then you will have a tracking number, which will show up in the order record.
I've shipped well over 4000 items with the USPS and only one ever was truly 'lost', some place in the Canada customs. I have not ever had a US package that was truly lost.
If your customer paid through Direct Checkout via Etsy (as opposed to Paypal) then Etsy will help you with the charge back (what happens when someone asks their credit card to cancel the sale). If you do not have a tracking # then there is nothing that can be done and you will probably need to refund the customer or let the CC company take the funds. I would update your policies with regard to this situation. If you refund the case without filing claims, etc, then your case is closed.