Below is the etymology, from Jeff Miller's Earliest Known Uses of Some of the Words of Mathematics (F). Perhaps a native French speaker can lend further insight.
FACTORIAL. The earlier term faculty was introduced around 1798 by Christian Kramp (1760-1826).
Factorial was coined (in French as factorielle) by Louis François Antoine Arbogast (1759-1803).
Kramp withdrew his term in favor of Arbogast's term. In the Preface, pp. xi-xii, of his "Éléments d'arithmétique universelle," Hansen, Cologne (1808), Kramp remarks:
...je leur avais donné le nom de facultés. Arbogast lui avait substitué la nomination plus nette et plus française de factorielles; j'ai reconnu l'avantage de cette nouvelle dénomination; et en adoptant son idée, je me suis félicité de pouvoir rendre hommage à la mémoire de mon ami. [...I've named them facultes. Arbogast has proposed the denomination factorial, clearer and more French. I've recognised the advantage of this new term, and adopting its philosophy I congratulate myself of paying homage to the memory of my friend.]