See this better introduction to the history:
Of course, the notion of "action" only becomes meaningful when one actually knows what the purpose of the action is - to be minimized. Around 1744 and 1746, Pierre Louis Maupertuis figured out that this could be a way to formulate laws of physics when he generalized Fermat's 17th century "principle of least time" (for the trajectory taken by light in any environment with a variable index of refraction) by this proverb:
Nature is thrifty in all its actions.
Obviously, some word had to be constructed or borrowed to describe the new quantity whose importance was previously unknown to the humans (and remains to be unknown to most humans even today).
Note that the word "actions" appeared as the only noun of the quote in the context of these minimization problems, so it became known as Wirkung ($W$) in German and action ($S$) in English. I am actually not sure why the letter $S$ was chosen.
There have been claims that Leibniz had found the principle as early as in 1707.
Maupertuis also wrote:
The laws of movement and of rest deduced from this principle being precisely the same as those observed in nature, we can admire the application of it to all phenomena. The movement of animals, the vegetative growth of plants ... are only its consequences; and the spectacle of the universe becomes so much the grander, so much more beautiful, the worthier of its Author, when one knows that a small number of laws, most wisely established, suffice for all movements.