I am not sure about this to be honest. Should an answer be selected anyway, regardless you feel your question hasn't been satisfied? How can you tell if people forgot to select the best answer or if they believe no answer was good enough?
Selecting best answer gives points, so you could say its in people's best interest to select one (at least to those hoping to get the fansite item one day). Even more, it seems that sometimes people may select best answer too soon (just like when people queueing to take a plane are desperate to get inside, even if the plane is gonna depart at the same time anyway - jk :).
There is also this tool that brings back to top old questions; maybe it could bring to top questions without best answer chosen (if not already), and send a reminder email to its creator at the same time.
I do think however that your proposal could be applied to questions asked anonymously.