The phrase suggests the effort taken to achieve something, not just the outcome. Despite the difficulties, she made a good fist of organizing the event.
Pro Tip 2/3
Subjectivity
What makes a 'good' or 'poor' fist can be subjective, varying by someone's perspective. I thought he made a poor fist of the negotiations, but others disagreed.
Pro Tip 3/3
Cultural Nuance
This phrase is more common in British English and may not be well understood elsewhere. In her first cricket match, she made a better fist of batting than bowlers expected.