The closer the POD is to the year 1000, the more time there is for changes to accumulate. Also, if the POD takes place in a place well connected to many other prominent places, the butterflies should be flapping harder; a POD in the Fertile Crescent is more likely to lead to large changes than one in Scandinavia. That is why the POD should either take place in Europe or a place with strong ties to that continent, because post-1500 it is European colonisation and trade that had the strongest global consequences, as opposed to regional ones.
Not sure how feasible this is and someone please correct me if what I write is implausible. Basil II, aware of the poor calibre of successors available in the Macedonian dynasty, decides to formally adopt and appoint as successor a young man who was the son of one of his old generals and who was already being raised at court (well, in a monastery but overseen by Basil), one Isaac Komnenos. If Basil dies when he did in OTL, Isaac I ascends the throne at age 18. Now, I'm gambling the prestige of the dynasty and Basil II himself makes people fall in line and that Basil's brother, OTL Constantine VIII, a purportedly indolent man, either does not rebel or does and fails (again, someone correct me if this is implausible). Isaac marries Catherine of Bulgaria as in OTL, which might bring the Bulgarians in line. He more or less continues Basil's policies and if he lives like in OTL, he'll rule for 35 years, leading to nearly a century of good emperorship. Now, perhaps Isaac will have a surviving son ITTL, perhaps he will be succeeded by his brother John and then an alt-Alexios I. Given OTL Komnenid fortune/skill in raising children/whatever, let's assume that at least in the half century after Isaac's death the emperors are somewhere between decent and great.
Meanwhile, the Seljuks have taken Persia and are becoming more Persian themselves. The Turkish migration keeps moving west and they raid enemy territory. Only, where in OTL they took advantage of Roman weakness in the 1040s and 1050s, here they hit a wall and fail to defeat Isaac. After a period of failed or inconclusive raids, the Seljuks decide to go for what is perhaps the even greater enemy and an easier target; not orthodox Rome is their goal, but shia Egypt. The Turks fall on Egypt like a storm and manage to end the Fatimid dynasty, after decades of struggle or perhaps if Alp Arslan is born despite the POD the Turks win within years. Turkish penetration in Eastern Anatolia is light to nonexistent, with the Turks settling down in Persia, Mesopotamia, Syria and lush Egypt.
With the Seljuk Empire stretching from Persia to Egypt, the rulers will have a difficult time keeping it all together. Perhaps things fall apart not long after the conquest of Egypt, perhaps it all remains united for decades or even a century, threatening the Romans. In any case, with the Komnenids at the helm I say that nothing like Manzikert happens in the 11th century at least and even if something like that were to happen, there wouldn't be enough itinerant Turks and Turkmen to demographically change Eastern Anatolia, much less the rest. Sooner or later, the Seljuk empire fragments and the successor states fight each other more than the Romans or others.
Now, why the focus on the Romans? Because their geographical position means their actions have direct consequences on three continents in the short term. An empire led by Isaac I immediately after Basil II is liable to hold the Danube border and take Sicily and perhaps the entirety of Southern Italy. This in turn puts the pope in a bind, potentially butterflying the schism or at least delaying it; Leo IX might very well never become pope. The pope, whoever it is, might suck up to Constantinople or might lean on the German emperor, butterflying the investiture controversy. Henry III tried to subdue Hungary more than once; perhaps in this world Aachen and Constantinople fight over who gets to have the Hungarians in their camp. Norman Sicily will not exist here, so Norman interest in England is likely to be even greater than it was IOTL or perhaps many Normans become Varangians and end up somewhere in the Middle East. In any case, the mess that is 11th century England could have a number of outcomes, among others an Anglosaxon England. Then, with the Turko-Persian Seljuks taking Egypt, that country would take on many Persianate characteristics, producing a different country compared to what we are used to. An earlier Fatimid demise means that the Banu Hilal never go west, irrevocably changing North Africa and quite possibly Iberia. A stronger Roman empire with a presence in Southern Italy and possibly Venice would influence the North Italians, who are in the midst of their Commercial revolution. A Roman Empire with stronger western ties and still in the Macedonian renaissance may very well spur on a quicker and more importantly stronger counterpart to the 12th century renaissance, causing all kinds of butterflies. Looking further afield, all the butterflies would prevent Genghis' birth and presumably the rise of the Mongols, stronger trade links might make one of the Greenlandic Norse think 'hey, maybe there is something to be found in that western land cousin Snorri can trade for the laterst Roman/Persian/Indian luxury', who knows what the Song might get up to if their dynasty gets to live out its natural life, hell, perhaps it's Berber North Africa which tries to circumnavigate Africa in order to cut out the Roman and Turkish middle man. By the 21st century, the world will be a very different one.