This is a remnant of the Ottoman "millet" system, which was an ethno-religious division of Ottoman society into discrete and self-government autonomous units. The Ottomans basically told every group of discreet people that they could govern themselves and handle their own internal community disputes according to their own traditions. In return, when the Ottoman government needed something from this or that millet, they turned to that millet's head to coordinate and micro-manage for them. Effectively, this minimized the impact of Ottoman-rule (which might have felt heavier otherwise) on peoples newly conquered. So Greeks were more or less autonomous under the Greek Orthodox Patriarchate, the Armenians under their Patriarchate, etc. This is where it gets a little tricky, however. The Greeks, for example, claimed that all Orthodox should be government by them (this would mean that Greeks could send high ranking officials from Istanbul's Phanariot community -greek-speaking orthodox elite) to govern Bulgarians, Serbians, Romanians, Armenians, etc. They did just did, but eventually some of these groups lobbied for their own unique millets. And the Ottomans often obliged (they didn't care as long as the system overall was stable).

Essentially, the Armenian millet in Jerusalem was big enough and important enough for local artisanship and trade that they won the privilege of having their own separate millet... with their own separate quarter of Jerusalem. In fact, overall, Armenians had two separately governed millets of their own (Armenian Orthodox and Armenian Catholic). That's because the foundation of a millet was confessional, based on religious practices. So if Armenians were split into two groups by church, then they were split into two groups by law, living arrangements, jobs, etc.

That being said, there have always been Armenians in Jerusalem, but it was under Ottoman rule that various religious differences between Christian groups coalesced into a separate quarter for Armenians.

The current walls (literally separating the city) were built, for example, in 1558 -- under the reign of Suleiman I, a Sultan known to be particularly beloved by Armenians -- probably because he let them rule themselves and recognized their autonomy under a separate millet against the wishes of Greek Christians who would have preferred to govern Armenians on behalf of the central state. Armenians were particularly beloved members of the Ottoman mosaic, being known as the Millet-i-Sadik (the Loyal People).

Bonus:

• Who governed the 'Christian Quarter"? They all share it.

• Who governed the 'Muslim Quarter" Overall the Sheikh ul Islam but locally the Imam of Al-Aqsa mosque. Good luck if you're shiah tho! lol. "Only Sunnis need apply".

In her book, Empire of Difference, Karen Barkey argues that it was this unique approach to managing differences (millets were based on 'difference' between each other) that allowed the Ottomans to leverage various local loyalties against each other and in concert with the central state for prolonged stability, thus helping us understand not why the Ottomans fell apart but why the Ottoman realm stayed together as it did for so long.

I’d also like info on the now gone Moroccan quarter(?), why was Morocco (in addition to Armenia) so prominent in the Old City?