I responded to Mike Scott's answer to prevent the 4th crusade by having Alexios IV Angelos die as a child with:
It would be better to have his uncle Alexios III Angelos die in infancy or young adulthood and never usurp the throne in the first place, thus giving Alexios IV no need to seek outsiders to overthrow Alexios III. That would eliminate the 4th crusade just as well and also eliminate 8 years of rule by Alexios III.
Note 01-13-2019. Alexios III is blamed for squandering the imperial treasury and leaving the government unable to find the money that Alexios IV promised the Crusaders, thus leading to their second siege of Constantinople and takeover.
What would be better still would be if Andronikos I gets executed for treason sometime in the reign of Manuel I and never usurps the throne in 1183, and thus is never overthrown by Isaac II Angelos in 1185 who thus is never overthrown by Alexios III Angelos in 1195, thus preventing the 4th crusade, the reign of terror of Andronikos, and 20 years of misrule by Angelos Emperors.
Added Jan 13, 2019 And the change could happen earlier for even better effects.
1) Prevent the defeat at the Battle of Manzikert in 1071, thus leaving the eastern Roman Empire much more powerful than it was at even the height of the Komnenos Dynasty about 1175, let alone in 1203/04, and thus a much less tempting target for the 4th crusade. And it should certainly be rich enough to pay the 4th crusaders everything that Alexios IV promised and send them on their way to the holy land.
2) Or Manzikert happens and also Alexios I Komnenos never becomes emperor in 1081. The Seljuk Turks continue to conquer Asia Minor and then begin to conquer lands in Europe about 250 years before the Ottomans did. The situation becomes desperate and the eastern emperors force the Patriarchs of Constantinople to end the schism with Rome in order to make getting help from western Europe easier.
3) Prevent the church reform movement in the west in the 10th and 11th centuries, since many reformers sought to make the Catholic Church and the Pope the secular ruler of Europe and the world. This would prevent crusades and power-hungry popes.
4) Keep Venice part of the eastern Roman Empire and too dependent to ever lead a crusade against Constantinople.
5) Prevent the Lombard invasion of Italy and the Slav invasions of the Balkans in the 6th and 7th centuries, thus leaving the eastern Roman Empire in firm control of Italy and Rome, and preventing the Patriarch of Rome from becoming any more powerful than the Patriarchs of Constantinople, Antioch, Jerusalem, or Alexandria.
6) Prevent the Arab-Islamic conquests in the 7th century, thus keeping the eastern Roman Empire larger and more powerful and less likely to be a target of western adventurers.
7) Prevent the great Plague in the reign of Justinian and/or the Ostrogothic resistance under Totila (reigned 541-552) that devastated Italy. Thus the Empire as a whole will be more powerful and able to hold onto Italy, and Italy will be much more desirable for the Empire to hold onto, and the Patriarchs of Rome will never get much political power or influence or cause any schisms with the Patriarchs of Constantinople.
I think that a combination of one or more of these should prevent the Catholic-Orthodox schism.