Full quotation is from the World History Encyclopedia (emphasis added):
The geographical extent of the Byzantine Empire changed over the centuries as the military successes and failures of individual emperors fluctuated. Territories which were held in the earlier part of the empire's history included Egypt, Syria, Jordan, Lebanon, and Palestine. Greece was less important in practical terms than it was as a symbol of the Byzantine's view of themselves as the true heirs of the Greco-Roman culture. Italy and Sicily had to be defended, ultimately unsuccessfully, against the ambitions of the Popes and the Normans. The Balkans up to the Danube River were important throughout, and Asia Minor up to the Black Sea coast in the north and Armenia in the east was a major source of wealth, but both these regions would require regular and vigorous defence against various perennial enemies.
The Byzantine entry links to an entry on ancient Armenia, but the latter focuses on kings and borders rather than on economy.
Hey, this is certainly an interesting question but you're likely going to have a hard time getting an answer, because you are misreading the above quote.
The quote is referencing two major regions and really ought to be read as so: (The Balkans up to the Danube River were important throughout,) and (Asia Minor up to the Black Sea coast in the north and Armenia in the east was a major source of wealth), but both these regions would require regular and vigorous defence against various perennial enemies.
The major source of wealth being referenced here is Asia Minor, which has it's north coast along the Black Sea and has Armenia on it's eastern border.
Asia Minor was a major source of both agricultural and resource wealth for the Byzantine Empire up until the fall of the Kommenos dynasty led to the permanent lost of the eastern imperial territories. It also provided a critical pool of manpower allowing the empire to sustain itself in major wars. The fact that the entrance to Asia Minor was bottlenecked by mountainous terrain in the south and Armenia (a traditional Byzantine ally) in the north allowed it to survive the initial onslaught of muslin conquests in the 700-800s. After other major territories we're lost Asia Minor remained and became a core part of what most people think of as Byzantium.
Hopefully that helps explain the situation. I think you're just misreading the quote and that is creating confusion.