Lingua Franca is of essential interest to creole studies for several reasons. For one thing, it is the oldest pidgin for which we have a decent amount of data. The first text in what is clearly Lingua Franca dates from 1353, but there is also material from the 13th century which may represent an earlier version of the language. It is quite likely that it had existed for some time by then, and it has even been suggested that the origins of the language lies in a simplified trade Latin used by Jewish traders.
Given its continued use into the 20th century, Lingua Franca also stands out as perhaps the most long-lived pidgin language we know of. Pidgins are usually rather transient phenomena which are either nativised and creolised, or vanish as contact between the groups involved ceases (or, paradoxical though it may seem, because contact gets more intense the use of Russenorsk, the Russo-Norwegian pidgin of the Arctic Ocean, declined precisely as many Russians and Norwegians established closer relations and instead learnt each other's languages.) Although it certainly changed over the centuries, Lingua Franca, however, remained severely reduced throughout its career, much to the joy of us pidginists. With an affix inventory consisting of one single bound morpheme, and with grammaticalisation of tense/aspect distinctions only in its incipient stages, it does not seem to have been either expanded or nativised anywhere.
As the use of Lingua Franca spread in the Mediterranean, dialectal fragmentation emerged, the main difference being more use of Italian and Provençal vocabulary in the Middle East, while Ibero-Romance lexical material dominated in the Maghreb. After France became the dominant power in the latter area in the 19th century, Algerian Lingua Franca was heavily gallicised (to the extent that locals are reported having believed that they spoke French when conversing in Lingua Franca with the Frenchmen, who in turn thought they were speaking Arabic), and this version of the language was spoken into the nineteen hundreds, witness Schuchardt. Holm's suggestion that it was this variety of Lingua Franca which through relexification developed into Algerian French seems somewhat far-fetched as can be seen from Lanly's study, Algerian French was indeed a dialect of French, although Lingua Franca certainly had had an influence on it. Nevertheless, there is no doubt that Lingua Franca did have an impact on Algerian French.
Lingua Franca also seems to have had an impact on other languages. Eritrean Pidgin Italian, for instance, displayed some remarkable similarities with it, in particular the use of Italian participles as past or perfective markers. It seems reasonable to assume that these similarities have been transmitted through Italian foreigner talk stereotypes.
It has also been suggested that the influence of Lingua Franca reached even further, and that it would be the ancestor of all the world's creole languages. Originally proposed by Whinnom, this radically monogenetic scenario claims that Lingua Franca was relexified into a Portuguese-lexicon pidgin, which in turn relexified into the creole languages that are spoken today in various parts of the world. The evidence that Whinnom and his followers presented was rather slim, however, and few creolists today would subscribe to the idea. Nevertheless, it is quite feasible that Lingua Franca had an impact on Portuguese foreigner talk habits, and thereby indirectly played a role in creole formation. There is also some evidence that suggests that Lingua Franca was indeed transported out of the Mediterranian, and Barbot frequently mentions its use in various locations in West Africa, and in a short 16th century word list from the Gold Coast, compiled by Spencer, the word molta 'much' looks more like Lingua Franca than Portuguese (though the word list is claimed to represent an indigenous language.) The explanation could be, however, that many of the Portuguese sailors who frequented the area were Genoese rather than Portuguese. Even from the Americas, some travellers' accounts occasionally comment on the use of Lingua Franca. A word of caution might not be amiss, though, for the term "Lingua Franca" need not have applied to the Mediterranean contact language, but perhaps rather to a local Portuguese-lexicon pidgin.
So, while the influence of Lingua Franca has been exaggerated by some authors, it seems clear that it indeed did have an impact on other languages in multicultural settings.
The most important legacy of the language's once dominant position in inter-ethnic communication in the Mediterranian is of course that its name has entered not only English, but also numerous other languages as a generic term for precisely that a language of inter-ethnic communication.
Both structurally and sociolinguistically, the characteristics of Lingua Franca agree well with what we see in other pidgin languages around the world. As already mentioned, the morphology is reduced to one single affix, and, as first pointed out by Schuchardt, verbs are normally in the infinitive form, despite this form's not being the most frequent one in spoken Romance. Including numerous other characteristically pidgin-like features, Lingua Franca must be seen as a rather good representative of the class of languages known as pidgins. Obviously, then, the study of Lingua Franca is of prime importance to pidgin and creole studies, but alas, most creolists (including myself, I'm afraid) only have a very superficial knowledge of the language and its fascinating history.