It's interesting that everyone here has suggested families of non-Arab background, despite Arabs being the bulk of the empire.
So far it's all either Balkan families or the Girays. For the former, it's because the Arabs had little of a presence in the high echelons of the empire, unlike Balkan peoples (correspondingly, the Arab lands especially outside Syria were governed much more loosely than the Balkans or Anatolia). Arabs being important central bureaucrats was a rare occurrence; Arab Receb Pasha, a commander-in-chief during the Great Turkish War, was one of the few who rose so high, and even he was prejudiced against. The users mentioning the Girays are doing so because many Ottoman statesmen believed the Girays to be a legitimate alternative to the House of Osman, since the Crimeans were by far the most important vassal of Istanbul, not only militarily with their light cavalry but in terms of legitimacy emanating from the Giray descent from Genghis Khan. But your timeframe makes it very unlikely, bordering on impossible, for the Girays to take the Ottoman throne; the Treaty of Kucuk Kaynarca when the Crimeans gained "independence" from Istanbul is in 1774.
Who were the most senior noble families in places like Syria and Mesopotamia?
In Egypt in 1770 the power broker was Ali Bey Bulut Kapan, who was a Georgian mamluk and not Arab as you wanted. He's not too different from Muhammad Ali Pasha, really; Ali Bey also attempted to create a virtually independent domain centered in Egypt, going so far as to ally with Russia and invade Syria (the Ottomans had to bribe his general to withdraw from Damascus, and the said general then overthrew Ali Bey and seized Egypt). Ali Bey himself was a mamluk of the most powerful family in Egypt, the Kazdaglis, who were unfortunately for our purposes Turkish or Greek in origin and not Arab.
In Syria, all the governors of Damascus between 1725 and 1783 belonged to the
al-Azm family, an ancient Sunni Arab lineage who served the rulers of Syria since before the Ottomans (a
member of the family was actually Prime Minister of Syria as late as 1963). But they were never seriously a threat, as their legitimacy was dependent on Istanbul's patronage as governors of Damascus, and the Ottomans more than often saw them as proxies enabling the government to continue to exert influence in Syria. Aleppo was virtually ruled by Janissaries. Other powerful Syrian families included the Ma'n and Shihab in Lebanon. Palestine was dominated by the Zaydanis until the government dismantled their realm in 1775. None of them had the resources necessary to become ruler of the entire empire, and with the exception of the Azms (who were viewed as central Ottoman elites and local Syrian elites at the same time) few of them had important connections in Istanbul.
Iraq was ruled by a
mamluk dynasty for most of the 18th century, but I don't know much about them. They weren't Arab anyways.