I think they should support both the "accepted" and "best" answer use cases. Have separate indicators in the UI for them.
In the common case, the accepted and best answers will probably be the same, so keep the classic green checkmark there.
When the accepted and best answers don't match, have two checkmarks (or whatever) - one that explicitly says "user XXX accepted this answer" and the other that says "this is the best answer".
They have this in the form of upvotes. If the community disagrees with the selected answer, they can use upvotes and downvotes to reflect their opinion. Then, a user who is reading the question can sort answers by vote count.
In the common case, the accepted and best answers will probably be the same, so keep the classic green checkmark there.
When the accepted and best answers don't match, have two checkmarks (or whatever) - one that explicitly says "user XXX accepted this answer" and the other that says "this is the best answer".