The biggest thing the Byzantines needed was stable succession. Not necessarily hereditary. Just peaceful transfers of power so that they aren't regularly gutting themselves with civil war and preferably not having all a good Emperor's accomplishments undone by a incompetent successor.
I've always thought that the best choice would be for them to take advantage of the fact that their succession wasn't necessarily hereditary and institute a kind of meritocratic system. Though I haven't figured out the particulars of how that would work.
Exactly. The problem would be getting rid of the idea that if you have an army, you can fight for the throne.They could always go back to the older Roman practice of adopting qualified successors.
With a POD no earlier than 1000 AD, make sure Eastern Roman Empire keeps Anatolia until modern day. What would this development change elsewhere in Europe, Middle East and the world?
"How" seems to be already answered more than once so I'll concentrate on the changes.
This change more or less implies an absence of the Ottoman Empire as we know it: if there is some analog, then its conquests are limited to Asia and Africa (and perhaps some islands on the Med).
The consequences are enormous:
1. No Ottomans on the Balkans, which means survival of the states like Bulgaria, Walachia, Moldavia, Serbia (in whatever modifications).
2. Hungary is never conquered by the Ottomans and may remain independent from the Hapsburgs. Anyway, most probably it remains an elective monarchy: the Hapsburgs became the hereditary rulers (1st time in Hungarian history, IIRC) as a "reward" for liberating the country from Ottomans.
3. The Crimean Khanate is much less powerful and can lose its independence earlier than in OTL (either to the Commonwealth or to the Russian state). Quite a few extra possibilities there including an earlier Russian conquest of "Novorossia" (areas to the North of peninsula) and access to the Black Sea.
4. No Ottoman Wars in Europe between XVI and late XIX century. Austria, Commonwealth, Russia have much fewer wars to fight with an open question: what are they going to do with their free time.
5. It is reasonable to assume that the "Issue of the Straits" is different from OTL if they are in the hands of the Christian Orthodox state.
6. Prince Eugene has a much lesser chance to make his military career by the time of the War of the Spanish Succession, which means that France has a very realistic chance to win it on land.
7. Most probably, no "Bosnian crisis" which shaped Russian reaction to the Austrian ultimatum to Serbia in 1914.
Also, Islam (presumably) never reaches the Balkans, so the religious picture there is simpler. OTOH, perhaps the Greeks become viewed as the great oppressors of the Slavs.
Also, Islam (presumably) never reaches the Balkans, so the religious picture there is simpler. OTOH, perhaps the Greeks become viewed as the great oppressors of the Slavs.
Plenty of ways to do this:
-Basil's Niece Zoe is born a Nephew: The result is ensuring another 30ish years of stability that would aid during the rather shaky years of the middle 11th century.
-The Seljuqs migrate into India instead, and the status quo of Fatimid/Byzantine/Abbasid Cold War continues until the Fatimids implode.
-John II doesn't die of a poisoned arrow, lives another 10-20 years and takes Konya, pushing the balance of power in Anatolia decisively toward the Byzantines (even more so than historically). Manuel follows this trend as the peace treaty that he signed during his ascension wouldn't be there, and mops up resistance after a few decades.
-George Manaikos manages the armies longer during his tenure of the 1050s.
-Charles of Anjou dies shortly after the Sicilian Vespers.
-The Epirotes take Constantinople and block Nicean expansion into Europe, letting them focus more on Asia where the reformed smallholder army was more than a match for the various Turcomen Ghazis.
-Manuel Palaiologos doesn't ascend the throne and focus imperial energies towards Europe (and you lessen the feudalization tendencies of the time period)