I've run across a number of different questions that may have entirely acceptable answers yet an answer is not selected either due to a user not returning to complete it, forgetting, or not thinking it was perhaps the answer.
It's one thing if the answers provided are not necessarily what was being sought but what if say 10 or 15 other users believe an answer to be the best answer. Essentially, a question has either been answered or it hasn't so shouldn't nearly all answered questions have a selected answer ultimately?
One can look it this in a number of ways. There's completeness which in the event of a question having many answers may help pinpoint the best answer. I realize up-votes essentially do this but an up-voted answer could also rise above the selected answer when in the OP's eyes, the answer he selected was the most successful despite others having found other answers useful for different reasons.
If a certain number of people can agree that a specific answer is confirmed to also work for them, why should it not be selected? Considering that besides what I mentioned, this would really only effect reputation, it is perhaps not a concern or desire for many.
I personally view SE sites as being a potentially more useful specific answer S earch E ngine DB as it is essentially evolving into and as such completeness is always the goal despite that paralleling an attempt to count to infinity.
Other restrictions could apply like a required question lifetime before answers could be voted on to give the OP a chance (1-2 months? or even 6 months?). Also perhaps a voting period that would expire if not achieved in a certain time after initiated.
The OP should retain the right to deselect an answer that was vote-selected in the event that it truly isn't the answer but it could re-open again in 6 months or however long for another vote at that time.
It could also be an entirely separate selected answer (like a blue check mark) for designating a community selected answer. Something like this would no longer be about completeness and aside from the potential benefit of this additional feature, it is really already taken care of by up-votes.
The benefit of having two separate selected answers would primarily relate back to reputation/profile stats but in many cases there is one answer that should stand out just a little more.
I can't imagine this would be terribly difficult to implement within the existing SE system so unless there is a strong reason not to, I can't see why some variation of a Vote to Answer system shouldn't be implemented. There's a lot of potential for good in it.