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.
"There are two ways to be fooled. One is to believe what isn't true; the other is to refuse to believe what is true.”
Søren Kierkegaard, Danish philosopher (1813 - 1855)