give it one's best shot
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.
The American Heritage® Idioms Dictionary Copyright © 2002, 2001, 1995 by Houghton Mifflin Harcourt Publishing Company. Published by Houghton Mifflin Harcourt Publishing Company.