I agree, that it is the Solomon Islands. I wasn't sure if they were considered a country, but after checking out
they did in fact become an independent country in 1978. The same web site also says they were named after King Solomon of the Bible. So if the question is correct in saying there is only one country in the world named after a biblical king, The Solomon Islands is the correct answer.
The first documented contact with Europeans was made by Alvaro de Mendana in 1568. He discovered alluvial gold and perhaps thinking he had found the Biblical King Solomon's mine, he named them Isles of Solomon.