Very hard to have the Empire recover at that point from any location, much less a thin mountain peninsula populated by (male) monks with little defenses, natural resources or population.
Thessaloniki is somewhat close (not close enough for Mount Athos to be *part of* the city, but close enough for it to have some significant daily religious/political role in the governing of a country based out of that city; e.g a Quasi-theocracy with the Patriarch based at the holy mountain.).
That city has a fairly good prospect for being a capital; for much of Byzantine history it was the "second city" after Constantinople, and in the 14th century it was one of the larger cities around (100k+), being fairly prosperous. It doesn't have the best strategic location, being threatened by Serbs and Bulgars to the North, South Italians/Normans to the West and Turks to the East, but given the right PoD it can either be the capital of a Byzantine remnant (eg surviving Despotate of Epirus ) or alternatively the capital of a new Greek state in the event of an Ottoman collapse post 1453 (and it would need to be a collapse, the Ottomans presence in Rumelia is too strong at this time to be undone easily).