As the challenge says make either The Moors or Andalucians, The Caroligian or a French empire which isn't the Bourbons and the Eastern Roman empire last as far as plausible. Meaning if you want to last till the 20ht century it's up to you how many butterflies you massacre.
Andalusians, have the Battle of Tours be a crushing Muslim victory, allowing for the invasion of France, and allowing there to be no foothold for a Reconquistia in Iberia.
For a Carolingian Empire, Maybe have a marriage between Irene of Athens and Charlemagne, with ideas of Primogeniture changing the succession law either during the later years of Charlemagne when he only has one son, or Loius The Pious taking a good idea after the Carolingian and Roman Empires split again.
For a French Empire, have Napoleon fully pursue a lasting peace, fully throwing his weight in for a Franco-Russian Alliance.
For the Eastern Roman Empire, have Khoshorow I Never take power, and put a nobody in his place. Giving enough time for the Roman armies to fortify their conquests. Also, maybe have the Kingdom of Soissons, a Roman Breakaway state in Northern Gaul, come out triumphant, and form a sort of "Northern Roman Empire", to give Justinian an ally.
If we're not going Justinian's Empire, I'd suggest having either a pod that prevents Islam, or have it so Yarmouk is a crushing victory that allows the borders to be stable.
Have Basil II place importance on ensuring the succession, or just avoid the Angelids and the Fourth Crusade. At the latest pod to save Eastern Rome, I'd suggest the Varna Crusade being successful.