The 'franca' of Lingua Franca do indeed refer to the same 'Francs' who gave their name to France and the French language. But as stated in the Wikipedia Article the Lingua Franca language used in the Ottoman Empire didn't had much relation to French itself. It was a creole language heavily influenced by Italian who dominated trade at that time. 'Francs' designated western people in general, not specifically French people. Lingua Franca then become 'more french' in some regions of Ottoman Empire were France was more dominant like Lebanon.