Sometimes you can guess. Britain would have scrambled for a Prime Minister if Churchill had died between the world wars, or been shot during the Boer War. And one alternate history had Lincoln not get shot at Ford Theater -- and his reputation was rather lower when he had to manage the Reconstruction as well as the war. Napoleon observed that his reputation would have been much higher had only a cannonball carried him off as he rode down the streets of Moscow.
But that precludes a lot. Suppose Prince Arthur had survived. It's one thing if he survived long enough that the future Henry VIII was married off to someone else by the time of his death; either, if she proved barren, for him to fudge up a different reason, or for a future in which the Church of England did not exist, and what would that have done to the Reformation. It's another if he actually lived to succeed or even just to father an heir. But the trick of that is the alternitacy of it turns on what would he have been like as king, or what his purely hypothetical son would have been. Would they have funded Henry Hudson? Would the English colonization of North America gone off? Or suppose Constantine had not executed his oldest son, Crispus, who had looked to be his successor? (An execution where historians often accept the ancient historians' account -- that his second wife had claimed Crispus had tried to rape her, he had executed him in a rage, and then he had learned the truth and executed her -- even though it's obviously the story of Phaedra, Hippolytus, and Theseus, slightly tweaked, because of the lack of alternative reasons.) But what would Crispus Augustus have been like? Constantius II was Arian, and Julian the Apostate tried to revive paganism, both with startling lack of long-term effect, so his religious influence might have been minimal, but what would his other effects have been?
Perhaps this is one reason why many of the best alternate histories are brief accounts by alternate historians speculating about what the world would be like if something that really did happen, happened. Because they are always funny and off the wall. Benedict Arnold as the second president would have made perfect sense if only he had died at Saratoga.