The medieval (Bagratid) Armenia likely needs to survive instead of being annexed by the Byzantines in 1045. Byzantium needs to get off its high horse about not wanting another Christian ruler near the equal of the Emperor, and realize Armenia is a very useful buffer state to the Muslims.
If there is a strong Armenian buffer state, then there is no Manzikert scenario so this actually works out better for them as well.
So we have a strong Armenian state in eastern Anatolia. To meet the requirements of the OP, this needs to eventually expand westward, likely when the Byzantines experience some kind of defeat in southern Anatolia or northern Syria. The end result being Armenia expanding into central Anatolia so that Armenian demographics improve over the long term.
Lots can happen. The medieval kingdom does not need to survive as an institution. As long as Anatolia does not become conquered by Muslims, there can be any number of Armenian states that rise and fall before the modern era. The end result would be a Greek state in Europe, western Anatolia and likely the northern Anatolian coast, and an Armenian state in central and eastern Anatolia with access to the Mediterranean and perhaps coastline on the Caspian.