Maybe have the Ottomans win the war of 1877-78 against Russia? Or have none of the great powers, above all Russia, launch the war of 1877-1878, perhaps because multiple great powers, like Austria, Britain, Germany, voice express opposition to Russia going to war in advance. The Ottomans suppress the revolts of that time - Bulgarian and Bosnia-Hercegovinian, and its bloody, but not suffering great power invasion, loss of territory to rebels, or expulsion of Muslim populations by successfully rebelled new Christian states (Muhacir refugees), later potential rebels (Armenians) never develop separatist ambitions or are thought of as a threat and the Ottoman state and ruling class is less paranoid when more secure?
It's one of the more charitable, optimistic possible interpretations of Ottoman future, I know, but the logic is pretty straightforward, less paranoia and less feeling of siege, less violence and less concern with purging potential separatist wedges and elements.
The more uncharitable interpretation is that keeping the empire big and diverse just gives them more peoples to perpetrate genocide on, and genocidal behaviors/responses were baked in by the 1870s as shown by the behaviors of various irregulars like bashi-bazouks in Bulgaria in 1875-76.
So if that's already too late, maybe you need to go back further, and prevent the success of the Greek revolution in the 1820s, perhaps, by giving the Sultan the good judgment to not fan the flames by executing the Patriarch or harming other major Church figures or symbols. Maybe the Greek klephts and secret societies and uprisers can be defeated or their effort peters out before Russian or western philhellenes catch too much of a sympathy bug. And lack of Greek success offers far less encouragement to other Balkan nationalities for the remainder of the century, and prompts much less paranoia by the Ottomans as well.
Of course maybe the Greeks and Turks of the 1820s and the period preceding it had a generational memory of the bloody Greek Orlov Revolt and its just as bloody suppression by the Turks in the 1770s, in which case we should remove the Russian fleet operations in the Eastern Mediterranean that inspired it.
At this point I could just keep going back ad absurdum and just never have the Ottoman Turks cross the straits into Europe and content themselves with Asia.
