To play the devil's advocate...I hope the Constantine XI choice is a joke.
Prior to becoming emperor, Konstantinos XI did manage to stabilise the Morea as Despotes and IIRC conquered the crusader duchy at Athens. In his situation, there really wasn't anything that any leader could have done; it's impressive that he did as much as he did. Then of course, when Konstantinopoulos fell, he could have fled to Italy or beyond, but he stayed and (supposedly) charged the breach, dying with his empire in a fashion that (if true) would be heroic. There's a reason he's the Greek ethnomartyr to this day.
... Not that I personally think that any of that should qualify him for the greatest emperor, but he did manage to gain lasting symbolic importance.