If you used Ebay's Simple Delivery, then they will have refused the buyer's claim if the item shows as delivered.  I am sure you understand the buyer's frustration - they have bought something and not received it and nobody will help them.

I know it is stressful to be threatened with court action but the law regarding safe delivery to a buyer, and a seller's responsibility, varies for businesses and private sellers.  

You may choose to seek legal advice, or hope that the buyer is not serious about a legal claim, or refund the buyer.