I imagine a scenario could arise where an extremely well-respected and high-ranking member of the Varangian Guard converts to the Orthodox faith and is somehow able to ingratiate himself into the inner circles of the imperial court. Matter of fact, if said Varangian is adopted by the outgoing emperor, perhaps the Varangian is elevated to co-emperor. A marriage into the imperial family could be useful as well.
There have been several unlikely Eastern Roman emperors that rose from obscurity to be crowned, including Justin I, Justinian, and most notably to me at least, Basil I, to name a few. A lot of these guys weren't Greek at all either.