This article describes more information about the "No Best Answer" option.

Sometimes there just isn't a good answer. So when a question goes to vote, No Best Answer appears as an option along with the list of answers.

When a question expires with only one answer, the question automatically goes to a vote, with No Best Answer as the second option.

If an answer is chosen as the best, the question becomes resolved. However, if the No Best Answer option is chosen as best, the question is automatically deleted and the asker of the question gets five points credited back to their Answers account.