Here's a (most likely) insane way to do it. Have say Valens defeats the Goths at Adrianople. Details are unimportant: the Gothic leadership is slain and there are thousands of imprisoned Goths. Instead of enslaving them or sending back beyond the Danube, Valens transports them (in chains) to be settled in Egypt to serve as soldiers-settlers to keep the native Egyptian population in check in addition to bolstering Roman defenses against raids by nomadic Arab tribes. The Goths soon recover their numbers, expand their demographic presence in Egypt and in the span of a century, serve as relatively loyal stewards of the Roman state until one day the Persians (or some other enemy) does enough damage that the now pseudo-Hellenized Goths rebel, perhaps in the pretense of installing a puppet Roman as Emperor. They secure their independence perhaps in alliance to the Persians or some other power and rule Egypt.
See, this seems a vastly more plausible approach than most, as the Goths were literally the other side of the world from Egypt.
Yes, well this Gothic Egypt would only be Gothic in name and perhaps language, but culturally Romano-Greek. At best they might stay Gothic as much as Crimean Goths did.
To take this thinking a bit further. Assuming this new Egypt consolidated, and weathered the storm of the Arabs, could they potentially control trade in the whole Med and maybe open a route to India?
1) The Crimean Goths aren't Goths? I'm sure they'd have been surprised since they were the ones who stayed put when the Visigoths and Ostrogoths went west.
2) That assumes we have the Arabs - or even some of the issues in the West. There are huge butterflies - from a possibility of not needing Justinians Restoration.
But assuming they do come to dominate the province (I find that unlikely, it seems more likely they'd be one of a few significant demographic groups in Egypt - Goths, Greeks, Egyptians, Arabs, Berbers?) but assuming they are the major group that supply manpower. If they dominate the trade routes (I would expect that more of the Greeks and Arabs, rather than Goths), I could see them setting themselves up in negotiated fortified trade posts - in fact, that sort of self-imposed isolation/seperation would likely characterise them.
What IS interesting however, is that if they maintain their Arian Faith (unlikely, but a cool option), is that the Arians control Alexandria!