I answer a lot of questions, and I get so annoyed when the person who made the question doesn't select an answer.
What I thought was that if the person who created the question has not selected an answer within a certain period of time whichever answer had the most upvotes would get their answer selected. If all answers have the same amount of upvotes (or didn't get upvoted/downvoted at all) then the question can go to a mod for them to select an answer.
(Each mod could get a set of tags and the tagged question can go to that mod.)
And, if the question owner doesn't want a Best Answer, they can state a reason why. If the staff approve of this, they won't select an answer.