I'm thinking that the weekend is not a good time to offer a bounty because people are less likely to be at work. So fewer people get to see the offer before it expires.
Probably various statistics about this can be found - and it all depends a bit on how you define best time. I have tried to make at least some queries using Data Explorer. (Notice that it is possible to change the site in the queries, so you can compare the stats with some other sites.) Since I am pretty sure that there are many users on this site with great SQL skills, I am making this post community wiki - feel free to add some other queries if you have something interesting to share.
- Percentage of bounties with accepted answer depending on the day when the bounty started (graph)
- Average number of answers depending on day when bounty was started (graph)
It might be useful to know that in SQL `DATEPART("weekday", date) returns number which corresponds to Sunday = 1, Monday = 2, ..., Friday = 6, Saturday = 7.