My starter for ten would be a core Empire essentially covering Justinian's Roman Empire, plus some more territories in coastal Spain and Gaul, and some favourable territorial adjustments in Mesopotamia and the Caucasus. Add to this a wider sphere of indirect rule and influence covering much of the Caucasus and Arabia (let's say some sort of significant loss of Iranian influence in the region) and extending into the regions of OTL Hungary, Romania and Ukraine to encompass Christianised Slavic and Turkic tribes who derive their legitimacy from their relationship with the Roman Emperor at Constantinople, and are kept in check by subsidies from him. Ditto for those areas of north Africa outside direct imperial rule.
Such a construct would not be stable and unchanging and would wax and wane according to the power of the larger states on the frontier that are unable to be simply dominated, notably Iran but to a lesser extent Francia, Axum and theoretical steppe constructs. It would nonetheless be considerably more powerful than OTL Byzantium ever was, and there's no reason why it shouldn't live a China-esque existence of basic continuity.
All of this, it goes without saying, pretty much relies on no Arab explosion, so in my book doesn't count as a proper "Byzantium" but rather a more successful "late Rome", though of course where these terms begin and end is endlessly debatable.