Generally, bounties are used for one of three purposes:
This question has not received enough attention.
The current answers do not contain enough detail.
Reward existing answer
One or more of the answers is exemplary and worthy of an additional bounty.
The last one doesn't benefit from additional time, so we won't consider it.
The other two reasons most often seen are, by design, seeking additional attention to the question and/or its answers — they're generally an offer of a reward to the answer that generates the most forward progress toward a solution.
The bounty is given away regardless of a valid answer being found or not.
Yes, this is by design. The bounty seeks to "Draw Attention" to the question, or to "Improve details" provided in the answers. Whether or not the puzzle is actually solved, the bounty has done its job — it's attracted attention to the question (it's been given "Featured" status on the front page for the duration of the bounty), and offered a reward to someone able to claim it. The reputation that it costs you as the bounty setter pays for getting attention to the question, not getting an answer to it; when it does both, that's a happy "coincidence", but just like an ad in the classified section (Wow. anyone remember those?), you pay for the listing whether it sells or not.
Even worse, this happens after a relatively short period already.
On this site, a week is an eternity. It's often 6½ days longer than a puzzle needs to be solved.
Having said that, it's not like the bounty can only be awarded to a successful solution; the purpose of the bounty is to attract attention, and if someone makes forward progress that's usually enough to keep enough forward momentum going to get the puzzle solved, even if that solution doesn't happen within the bounty window. If, even with more attention during a bounty window, the puzzle doesn't see forward progress, that's a good sign that what's missing is not attention to the puzzle or motivation to the solvers (you've just supplied both), it's a lack of a thread to pull at to start toward a solution path—you probably don't need a bounty, you probably need a well placed hint or two.
So it's not a matter of needing more time to get noticed, or more time for someone to have an epiphany. If the right person hasn't had the right idea in a week (or perhaps just hasn't seen the right award amount to make it worth their while), then extending the bounty time isn't going to make any difference.
Finally, having a puzzle bounty last more than a week would make the bounty lose significance to the audience; it just becomes background noise. Glorfindel's suggestion would be a way to re-up the ante, providing fresher and greater visibility and incentive, if you think that's all that's missing.
If you're hoping to make a single bounty investment in a question to push it to getting a solution, don't be disappointed if that doesn't happen.
- That's not what bounties are for—they're for getting attention, not an answer, to the question
- Often, a bounty is the wrong thing anyway; bounties attract eyes, but hints provide forward progress.
Parting thought: there's currently efforts in place to unify, not keep adding special cases for, various sites on the network so the code base is easier to maintain and extend. Making special case options for any single site is unlikely in general; given the current climate, the likelihood is vanishingly small.