I posted a question on Christmas and placed a bounty on it. I'm planning to re-bounty the question for 100 points today, but the system isn't letting me do that. Also, it is saying "award this bounty to the best eligible answer as soon as possible" when there is no answer. Is the website supposed to work this way? If so, why?
Regarding the site saying to award the bounty as soon as possible, even though there's currently no non-deleted answer, I suspect that's because there might still be a new answer that's posted before the bounty expires. In particular, it's a reminder for you to check for, and be aware of, any last minute answer that you might consider appropriate to award the bounty to.
Note that since your original bounty was not manually awarded, it would only expire when the next check for auto-awarding runs after the grace period ends. The network Meta's Time before bounty awarded automatically post has animuson's answer which states:
The check for bounties runs once an hour. Just looking at data here, all automatic awards seem to take between 1 and 2 hours from when the bounty ended to occur. I honestly don't know why there's always an extra hour delay; there must be an explicit check that is an hour off somewhere (as in the script that checks eligibility for auto-award seems to be looking for questions that have had the bounty an hour longer than when the grace period actually runs out).
Thus, the $68$ minutes in your comment of
I started the second bounty, but it appears that the first bounty expired 68 minutes late.
is consistent with the $1$ to $2$ hours timeframe mentioned in animuson's answer. This likely means the problem, that was there in $2017$, of at least an extra hour delay hasn't yet been fixed. Considering how relatively minor this issue is, I'm not surprised it hasn't yet been checked into and taken care of.