For example, are there any foods, goods, language dialects, etc. modern to the region today, that can be attributed to the French occupation of Egypt (and Syria) from 1798 -1801?
Yes it absolutely did !
Before I start with my post, here are some reserves : my English is a bit rusty, so I apologize for the unintelligibility that might happen, and more, I will use almost exclusively French sources on the matter. Besides, I am only familiar with French categories of modern (late XV° century - 1789/1792) and contemporary (1789/92 - current). Though there are subdivisions, I hope this won't be a problem for understanding.
I don't know of any goods or food items that may have been introduced by the French expedition force, but there is something that you question haven't mentioned, and which is extremely important in order to analyze nationalisms and patriotisms in the Middle East : political ideologies and concepts.
Many precisions are needed before we take on the core of this answers. Prior to the the Egypt expedition, orientalism already had moments of popularity in France ; there was a "premodern" and a modern orientalism, which had quite an influence on French society, notable among intellectuals. The Persian Letters of Montesquieu are in this way remarkable : though they adress French citizens regarding their institutions and political life, the romanticized point of view of Persian citizens is a good illustration of the interest, if not fascination, French Enlightment writers had for what was perceived as "the East". This is not anecdotical, as it helped the growth of an idea of Arabs being under the shackle of foreign powers, mostly the Ottoman Empire (Filiu, 2015).
Then came Napoléon. His expedition had an interesting aspect : among soldiers and equipment came 177 scientists from various disciplines, and printing presses with arabs signs (such models already existed, but were almost exclusively used by Arab christians) ; both were used for propaganda issues. After his landing in ALexandria, Bonaparte multiplies islamophiles and arabophiles statements, explaining that his only enemies are the mamlouk, as foreign oppressor to an indigeneous people. In this way, he claims to free the Arabs from the yoke of the Ottomans, symbolized by the Mamlouks (Laurens, 1997). For the record, the local population didn't quite buy it, and the French expedition force had to commit its lot of repressions in the captured cities.
Anyway, the Expedition failed, and its primary goal (to disrupt the Indian road) failed also. Napoleon reembarked quite hastily, and with the death of Kléber died the dreams of a French Egypt. However, this military expeditions had a tremendous effect on the Middle East. By kicking the nest that was the Ottoman Empire, it paved the way for a decisive intellectual movement : the Nahda.
Nahda can be translated as 'renaissance', and was described as such by its actors (Mervin, 216). It's a fascinating but huge topic, so we won't be able to get too much into details, but I cannot encourage you enough to read about it. A political, litterary, religious and intellectual movement, it aimed at adressing two challenges : ottoman domination and western expansion (Filiu, 2015).
The Nahda had specifically eclosed during Mehemet Ali, khédive (something along a governor/vice-king) of Egypt from 1804 to 1849. A political and social modernizer, he builds in Egypt a modern state, influenced by the western models. Flanked by French advisors, he presents himself as the "successor of the effort of civilisation started by Bonaparte" (Laurens, 1997).
In 1826, he sends a "student mission" of 44 students from his court to study 'Frankish arts' in France. ALong these students are an imam from al-Azhar, Rifaa Tahtawi, who publishes a book (The Gold of Paris), in which he describes the state of art of western sciences, and above all, complains about a lack of development of these disciplines in the islamic world. This lack of development is not, in his vision, because of a general alck of intellectual resources among the islamic world, but because of the oppression of the Ottoman Empire. As a imam, when he calls for a political change, he then refers more often to Arabs than to muslims, and the political change that is wished often targets the Empire (Filiu, 2015).
Ali's Egypt develops itself as a state based on modern techniques and institutions, brought directly from the West. Along with these technical changes have been brought up two major ideas that were en vogue in Europe : the principe of reason, and the participation of the people in politics.
This renaissance has given life to many movements, that have since shaped the intellectual and political frame of the (mostly Arab) Middle East, from arab nationalisms to islamisms. [[please note that what we describe today as political islamism is totally different from the XIX° century islamism, which was divided in currents, but in general focused on the idea of political participation of muslims against the oppressions of the Turks and the West]].
TL;DR : I don't know for food, but the Egyptian expedition left a tremendous impact in Egypt, and the broader Arab world, by creating the opportunity for a generation of intellectuals to rise, which would be influenced by Western ideas and that directly shaped Middle Eastern for the following years.
Principal Sources :
Jean-Pierre FILIU, Les Arabes, leur destin et le nôtre, 2015 Sabrina MERVIN, Histoire de l'islam, 2016 Henry LAURENS, L'expédition d'Egypte, 1997
I think it sort of depends on what you mean, because I would emphasize the cultural/historical/political legacy and influence of Napoleon's invasion of Egypt rather than the more material examples that you gave which, after all, would be kind of surprising if there were a lot of after only three years of occupation.
Off the top of my head, the best example I can think of is L'Institute d'Egypte, which was founded by Napoleon in Cairo in 1798 (and which tragically was largely burned in a fire in 2011.) But to give an example of what I mean by a wider legacy and influence, L'Instute d'Egypte was only one small part of Napoleon's scientific expedition to Egypt. Between the publication of Description de l'Egypte and the discovery by French troops of the Rosetta Stone, we can't really point to either of those things as being some sort of concrete material thing that we can go to Egypt and point to today, but they basically triggered Egyptomania and the explosion of interest in Egyptology in the 19th and 20th century.
Any effort to answer the question "How important was [event X]" is basically an exercise in counterfactual history, because it presumes to be able to answer the question of what would have happened had event X not occurred. So while I don't think we can say that there wouldn't have been any European interest in Egypt absent Napoleon's invasion (after all, the Pyramids weren't going anywhere), it seems to me that you can actually draw a line of interest going back to the discovery of the Rosetta Stone and the publication of Description de l'Egypte that leads all the way to contemporary Egyptian tourism and Egyptology, such that, for example, even if Napoleon didn't found the Egyptian Museum it seems more difficult to imagine that museum being founded or Egyptology taking the course that it did without Napoleon's invasion.
There are many, many other such examples that could be pointed to. But we should also be wary of overemphasizing the importance of some developments that seem like major "firsts." So for instance, Napoleon is credited with introducing the first Arabic-language printing press to Egypt and the Arab world. But it's not as if that in-and-of-itself somehow led to some massive cultural revolution the way that it did in Europe. Successive regimes would also use a printing press but it would remain under incredibly tight state control, just as the Ottomans had succeeding in doing with a Turkish-language press.
Politically, I think the greatest influence of the invasion was the subsequent political vacuum that allowed for the rise of Muhammad Ali Pasha, who was arguably the most influential ruler in Egypt's modern history, with the possible exception of Nasser. But it seems odd to credit the French for their own defeat.
In sum, it can (and has) been argued that Napoleon's invasion of Egypt marks the start of "modern" history in the Middle East. But that's not to say that Napoleon showed up with ships full of "modern stuff", words, cuisine, goods, whatever, and then they just had it. It's more that it was a really dramatic turning point in the country's history that marked the start of a long and continuing interaction between Egypt and Europe that we can trace the effects of down to today.