give it one's best shot, to

Make one's hardest or most enthusiastic attempt, as in I don't know if I can do it, but I'll give it my best shot. This expression employs shot in the sense of "attempt," a usage dating from the mid-1700s.

To try one’s hardest; to expend optimum effort. Originally a military term, best shot in the sixteenth century denoted the soldiers who could most accurately shoot the enemy, according to William Safire. However, the word shot also had meant an attempt or a try from the mid-eighteenth century on, casting some doubt on this and other etymologies, which trace the term to billiards and boxing. In any event, in the twentieth century it became commonly used in politics (“The candidate was willing to give it her best shot”), as well as in other contexts. David Baldacci had it in Hour Game (2004): “‘If you can wake them up. . . .’—‘We’ll give it our best shot,’ said King.”