I think this would go a long way towards encouraging question acceptance, especially if it were the default sorting method.
Take this user, for instance. He has (at the time of writing) a 54% acceptance rate - 73 out of 135 of his questions have an accepted answer. How can he improve this acceptance rate? Currently, there's only 2 things he can do:
- Pick a sorting option from the ones currently available and work his way through each page looking for ones he hasn't accepted.
- Use the advanced search options as a work around,
user:me hasaccepted:0in the search terms would do it.
The problem with #1 is that it takes a lot of work when you have a lot of questions. It might be something that you start doing and bail out on when you realize how tedious it is. Later when you come back to it how do you pick up where you left off? You're not likely to remember the sorting method/page number. If his questions were sorted by unaccepted, whenever he comes back to his profile he can continue easily, nipping away at the unaccepted questions bit by bit until his accept rate has improved enough.
As for #2, not everyone knows about the advanced search options (they're limited to ninjas only).
A sorting method in the user profile may also serve as a reminder for these users to accept answers to their questions, something that's sorely needed for some users.
Of course, this also comes with a couple of issues - "Unaccepted" is more of a filter than a sort, but a filter is a little out of context amongst sort options. As a solution to this, I would suggest that the sort would be on "Date accepted" descending with unaccepted at the top, followed by most recently accepted. A sub-sort would also be required, for which I would suggest total answers, then date posted.
It may also be worth displaying the accept rate, possibly next to the "x Questions" heading.
* (yes, his freehand circle puts mine to shame)