Currently, my biggest problem with bounties is that most of the time you have to wait for the bounty to nearly expire before anyone pays any attention to it. On a question I had a bounty on recently, It wasn't until a few hours before the expiration that I started to get answers, and it didn't get a good one before the bounty expired.
This has been my experience nearly every time I've started a bounty: an initial small rush of activity, followed by a week of inactivity, followed by a rush of activity as the bounty is about to end.
On the featured tab, questions are listed in descending order of expiry time. Most people will only see those that are about to expire. You can filter by tag, but that's the limit of filtering available to you, and I'm guessing not a lot of people do it. I propose that the featured tab questions be reordered to try and alleviate this problem:
Randomly - as
/reviewis. This would mean all featured questions get equal exposure. Or...
Interesting Questions First - as the home page is. A user sees bounty questions they are likely to be interested in, and able to answer. This ensures that your featured question is seen by people who are likely to be able to help. Or...
Something Else - there may be a better way. Suggestions?
Note: This only really applies to SO, where the number of featured questions is very large. Most sites only have a handful at a time so the current system is fine.