What it says on the tin: At some point, a Christian dynasty, a branch of the royal dynasty of Ethiopia that claims direct descent from King Solomon, comes to power in Egypt.
I feel like it is more likely that Solomon himself or a (less crappy) Rehoboam takes over Egypt, but in lieu of that, maybe have the Muslim conquests sort of fall apart soon after Muhammad's death, meaning that Christians are still in control of Egypt.
Sorry, but that would take the intervention of ASBs. There's little proof that the Kingdom of Solomon was anything like as large as the Bible claims. There's no proof in terms of inscriptions or coins or buildings for anything outside Jerusalem and the surrounding hilly region. Plus the major players of the time don't mention Israel being a power in the region. Egypt was in a serious decline, but was internally fragmented, with those fragments still being powerful.
First the Solomonids need to conquer Nubia. That's the "easy" part. Then they need to conquer Egypt. Up until the 12th century or so Egypt has a Coptic majority, meaning the Ethiopians will have significant support amongst the locals.
But conquering Egypt from the Arabs will be extremely difficult for obvious reasons, but possibly doable if the Arabs are weakened enough by internal struggles. The problem is Ethiopia had significant internal struggles of its own in the era when Nubia and Egypt were most vulnerable.
The Nubian Kingdom of Makuria was actually Melkite (Local name for Chalcedonian Orthodox) for a long time, and was no pushover.
Yeah--Ethiopia won't find it easy to defeat Nubia, although maybe they can do what the migrating tribes of the 14th/15th century succeeded in. I'm pretty sure there was a Coptic minority in Nubia in that time, so it won't be totally hell to hold the place down. But compared to taking one of the centers of Arab culture and political power, it'll be a cakewalk.