Nothing to do with French seems a bit strong. It's related. From Brittanica:
> lingua franca, (Italian: “Frankish language”) language used as a means of communication between populations speaking vernaculars that are not mutually intelligible. The term was first used during the Middle Ages to describe a French- and Italian-based jargon, or pidgin, that was developed by Crusaders and traders in the eastern Mediterranean and characterized by the invariant forms of its nouns, verbs, and adjectives. These changes have been interpreted as simplifications of the Romance languages.
> lingua franca, (Italian: “Frankish language”) language used as a means of communication between populations speaking vernaculars that are not mutually intelligible. The term was first used during the Middle Ages to describe a French- and Italian-based jargon, or pidgin, that was developed by Crusaders and traders in the eastern Mediterranean and characterized by the invariant forms of its nouns, verbs, and adjectives. These changes have been interpreted as simplifications of the Romance languages.