To understand what the Lingua Franca was, you kind of have to understand the cultural nature of the Ottoman empire. The Ottomans called Europeans "Franks". The Ottomans controlled all of the lower Mediterranean (Algeria, Libya, Egypt, Palestine, etc). Now, you might expect the common language to adopted to be Turkish, or maybe Arabic, but this is not what happened. The Turks were very isolated in their rule. They would just send a few tax collectors around and soldiers, but they made no effort to culturally rule their subjects, who were a mixture of local people, Arabs, Greeks and others, all speaking different languages. The French, however, adopted a completely different policy which was to educate anyone they came into contact with. You may have heard of the "Knights Hospitaller", a Christian order. These men set up hospitals and schools everywhere. There were many other such organizations. So, what this meant is that in all the cities of the Ottoman empire were little groups of French making schools and it was common for the rich or educated, even though they were Muslim to attend these schools.
The result of all this is that the French language became the basis of a polyglot throughout the Ottoman empire which was combined with Greek, Arabic etc. This common language became known as the "Lingua Franca".