I would wait until the package is actually returned to you before issuing any kind of refund. I've had this happen once and when the package went back to the sort facility it got kicked back to the correct place and was eventually delivered to my customer. If it goes back and forth more than once you'll need to call USPS to get it out of their system.
I'm not sure how this is your fault as the shop owner. I've accidentally typed my address incorrectly when ordering online and when the shop contacted me about the returned item, I was asked to pay for a second round of shipping which I gladly did since it was my faulty address.
I wouldn't do anything until you've received the package back in the mail. You might want to create a claim with the USPS for delayed or lost mail, just to get the ball rolling and feel like you're doing something to appease the situation. Sometimes that magically makes the package arrive.
First class shipping does not have insurance so it's unlikely you would receive monetary compensation or a refund for the postage you paid.
When it is returned to you, I would ask the customer if they'd like you to reship the order (at your or their expense, depending on how you feel about that). If it was a 'made to order' or custom order I wouldn't personally offer a refund. If you do choose to refund, then I don't believe you'd need to lose the shipping fee which you've already paid, so think about how you feel about refunding, minus the shipping cost. It's only a few dollars, so that might be a nice customer service gesture.
Now might be a good time to double check your Policies. My policies state that there are no exchanges or returns and shipping charges are non-refundable.