The Merriam-Webster Dictionary says that the phrase “make a good fist of it” is British, and means to do (something) well. For example, “Despite her inexperience, she made a remarkably good fist of chairing the meeting.”
On the other hand, if you make a poor fist of a task, you are incompetent or inadequate at doing it. In such various forms, this phrase dates to the early 1800s.
The source is uncertain. The Oxford English Dictionary says it was a US expression. But the English Dialect Dictionary of a century ago noted it was northern English dialect.
In these expressions, “fist” is used figuratively for “hand,” as a cowboy would be described as a hand, or where you might give somebody a hand.
The Oxford English Dictionary cites Caroline Gilman’s Recollections of a Southern Matron (1838): “He reckoned he should make a better fist at farming than edicating.” It also notes a book by William D Howells, The Undiscovered Country (1880): “Mrs. Burton is really making a very pretty fist at a salon.”
A variation of the phrase is “write a good fist,” which generally means to write legibly or clearly or both. Robert Louis Stevenson, in The Ebb Tide (1894) writes: “‘I’ve a good mind to read you my letter,’ said he. ‘I’ve a good fist with a pen when I choose’.” Another variation has to do with Morse code. From Railway Signaling and Communications (1953): “Students must possess a knowledge of the elementary principles of electricity and a good ‘fist’ on the Morse key.”
And, just to add to the confusion, “fist” is sometimes used by itself to mean a poor attempt or a failure. To say that “he made a fist of that job” means he made a complete hash of it. The Oxford English Dictionary gives an example from The Life and Adventures of Dr Dodimus Duckworth, published in 1833 by the American writer Asa Greene.
I’ve occasionally seen the phrase used in British fiction, but have only heard it spoken in North America once or twice.
I only hope I’ve made a good fist of writing about it.