As The-Cat-o-Nine-Tales said in the feature focus thread:

"Since I shared it on CivFanatics when they asked, let me add this information here as well, a partial list of working titles for the generic quarters.

• Farmers Quarter
• Trades Quarter

• Market Quarter

• Research Quarter

• Commons Quarter

• Defense Quarter

• Harbour

• and a few others we are not ready to tell you about yet.

Feel free to go wild with your speculation. :) "

I figured I'd use this as a chance to go wild with my speculation. 

Of the generic districts, we see some dedicated to food, industry (trades quarter means craftsmen quarter, not trading quarter), money, and science, along with naval and land military quarters and the commons, which is likely for housing. There is also the city center and outpost quarters.

I would guess that there would also be quarters dedicated to the remaining yields of influence and stability. Based on the released emblematic quarters like the Confucian School and the Egyptian Pyramids, I would guess this would be a cultural quarter for influence and a monumental quarter for stability. 

Based on the EQs we have seen for the released cultures so far, it also looks like the orientation of a faction does not entirely determine its quarter. So while the Harappans have an agrarian orientation and the canal network EQ, which I assume is a higher yield farmers quarter, the Nubians have a merchant orientation and Meroe pyramids as an EQ, which I would guess to be a monumental quarter and not a market quarter. The Greeks, meanwhile, have a scientific orientation and the amphitheatron as an EQ, which I would assume is a cultural quarter not a research quarter. 

This could indicate that EQs will either not necessarily provide bonuses dedicated towards the faction's orientation (so the amphitheatron could give stability and influence, ignoring science) or that they would have additional abilities dedicated towards the orientation (so the amphitheatron could provide the standard influence of a cultural quarter, and also increase influence and science yields by x% when a city is happy). 

I also don't know how the base yield of a tile will change or not change the yields of a quarter. Would the output of a farmers quarter be a set amount of food plus the normal FIMSI yields of a tile, or would it be a multiplier times the normal food output of a tile? If quarters were just a multiplier of preexisting yields, farmers quarters would naturally be placed on tiles with food like flatlands or rivers, while trades quarters would naturally be placed on hills or forests. Research, market, and cultural quarters would either center around points of interest that give special yields, like strategic and luxury resources, or would provide their own flat yields.

What do you think of these assumptions? If they are true, how would you feel about this system of city development? What are some ways that it could improve on the Civ 6 district system, and some features that you hope they borrow from the Civ district system?

Assuming that EQs would function as a generic quarter while providing some additional benefit to cities they are built in that goes towards the orientation of a faction, here are my guesses of what the released EQs do based on their descriptions. Feel free to give your own guesses of what they could do or what quarter they could be.

Assyrians - Expansionists - Dunnu, a border fort built in conquered territories - Defense Quarter, gives standard defense plus x stability in conquered territories.

Babylonians - Scientists - Astronomy House, temple home to priests, astronomers and mathemeticians - Research Quarter, gives standard science plus x science per adjacent cultural quarter or city center

Egyptians - Builders - Egyptian Pyramids, eternal home of dead pharohs - Monumental Quarter, standard stability, plus x influence, and plus x industry to trades quarters up to 3 tiles away

Harappans - Agrarian - Canal Network, farmers worked fertile land around Indus River - Farmers Quarter, standard food plus x food to farmers quarters adjacent to river or lake tiles in this region

Hittites - Militarist - Awari, outpost that could be adapted to the terrain - Outpost, standard defense plus x% defense from terrain bonuses

Mycenaens - Militarist - Cyclopean Fortress, large stones stacked without mortar - Defense Quarter, standard defense plus x defense per adjacent trades quarter

Nubians - Merchants - Meroe Pyramids, ruins of a rich and powerful civilization - Monumental Quarter, standard stability plus x money to merchant quarters up to 3 tiles away

Olmecs - Aesthete - Olmec Head, uncertain purpose but possibly represents fallen leaders or sacrificed enemies - Monumental Quarter, standard stability plus x% influence per defense levels of city

Phoenicians - Merchants - Haven, built in bays where coastal escarpments protected ports - Harbour, standard defense plus x defense and plus x money per adjacent land tile

Zhou - Aesthete - Confucian School, taught lessons on morality, justice, and how to be a righteous leader - Cultural Quarter, standard influence plus x stability and plus x% influence in city when happy

Aksumites - Merchants - Great Obelisk, graves for kings that were some of the tallest monoliths ever built - Monumental Quarter, standard stability plus x money per adjacent commons quarter or city center

Carthaginians - Merchants - Cothon, large harbour that housed and protected many ships - Harbour, standard defense plus x trade route capacity and prevents sea trade routes in region from being pillaged

Goths - Militarists - Tumulus, mounds of dirt and stone circles used to bury dead - Monumental Quarter, standard stability plus x influence when a unit from that city dies

Greeks - Scientist - Amphitheatron, devoted to performances of songs and plays for god Dionysus - Cultural Quarter, standard influence plus x% science and influence in city when happy

Huns - Militarists - Ordu, nomadic camp and origin of word horde - Commons Quarter, standard housing but able to move population to new tiles and regions by turning into tent caravan and resettling

All of these are speculative of course, especially the Ordu