I recently offered a bounty for a question. I have received only one reply to which I have given an upvote as the answer tries to solve the problem, but just falls short(as of now). Bounty is active for 24 hours more. So should I go ahead and award for an earnest attempt? I am asking this since the answer has only 1 vote as of now. Will System grant it if there are 2 or more votes ?
After reading your question, i think your answer is on the bounty information page.
When does a bounty expire?
Bounties expire after seven days. You will receive several notifications a few days before this happens.
If you do not award the bounty within 24 hours of the bounty period ending, half the bounty value will be automatically awarded to the top voted answer posted after the bounty start, provided it has a score of at least 2. If no new answer matches this requirement, no reputation will be awarded at all, and the reputation used on the bounty will be lost forever.
Please note that once a bounty is started, the reputation is non-refundable under any circumstances.
Addition from this answer.
Award the bounty to whichever answer is most useful to you. That's the idea. Just keep in mind that you won't get any points if you award it to yourself.