That's really difficult. I say the best chance is to have imperial marriage, best a series of marriages, cementing a strong dynastical claim of either the Western or Eastern Emperor on its counterpart. Then have a war of succession which ends in the Emperor ruling both.
Now if this is difficult, keeping both parts together and truly uniting them is even more difficult. If the HRE stays in Constantinople, we can take as granted that some counter-Emperor and the pope will cause trouble in Germany - and if the Emperor is in Germany or Italy, there'll be some Byzantine general or noble trying to go for the throne.
I'd say that several preliminary conditions have to be fulfilled:
- HRE should be by far more centralized, allowing an absent Emperor to hold it safe.
- Italy is the link between the Empires. So the whole of Italy should be firmly in the hands of the Emperor (may be HRE or Byzantine or both).
- that quite likely implies no pope in Italy or a pope firmly in control of the Emperor or no pope altogether, but merely the eastern patriarchical system throughout christendom.
- weak exterior enemies. In particular, no Avars, no Hungarians, no French no Ottomans, no Arabs, no Vikings being a truly lethal thread to either part of the Empire. Even if they are not powerful, they'll surely ally themselves against the Imperial behemoth.
I already read timelines for most of these points in this board - but never a timeline which required all of these...