Make a fist of something

This expression can take multiple forms because the word fist is often qualified by an adjective. For example, one can make a good, bad, brave, poor fist of something. There is some controversy whether the origin is British or American, but there is general agreement that it dates from the early 19th century. When one works at something with one's hands, the fist is generally involved in gripping or turning something with force, and hence is instrumental in accomplishing whatever task is at hand. In this expression, fist simply means doing one's best at doing something or other.