It could be like a mix of PoTD/HOH, and housing.
Gatherers and crafters could provide stuff for the dungeon like with housing. Crafters could e.g. create traps like in Halatali (Hard) or as seen in some of the Stormblood dungeons along with other interior. If a version of beastmaster is added at some point, one could place specific monsters/bosses in the dungeon if monster taming is something they could do, similar to FSH and aquariums.
Sizes
The sizes of the dungeon could be 1-5 floors? Maybe one could expand it from 1-2 floors and up? One could set up a biggish room area like in HOH and in trials, or several small with monsters, traps, puzzles, and boss(es).
Interior and enemy settings
The creator(s) could set and lock key terms to alter the dungeon floor(s) like with the Arcane Labyrinth in Final Fantasy II Anniversary Edition, and with that also set status effects like with POTD/HOH. Maybe one could place puzzles and set underwater movement in some of the areas/floors, along with furniture and interior pieces from the housing system. Setting i-/levels syncing could be done? Running through it with squad members or Trusts for testing could also be something to consider but would probably be too much work for the AI development team.
Roleplay
Roleplayers and FCs could customize and organize small dungeon crawls and events. With the world visit system one could arrange a larger cross-world customized dungeon crawl. For annual events like Halloween/All Saint's Wake, maybe a FC would like to create their own similar events.
PVP
Would that be something one would want to be able to toggle on/off in the player created dungeon? While still having monsters?
Rewards
The owner(s) of the dungeon could place a reward in a treasure chest if they wanted, like a cookie, or some other tradeable item or equipment piece. Maybe one could place a retainer at the entrance to resupply the treasure chest and greet challengers. The reward left by the creator(s) should then be visible? The rewards could also be something like with guildhests or levequests. In exchange, the owner of the dungeon could have the retainer collect fragments/other (untradeable?) items that "drop" in the dungeon. People challenging the dungeon could leave a guest book message or like if they wanted. Any other suggestions?
RMT and bots
A bit unsure but maybe have no XP and gil earning so bots can't use it for leveling or suck up the rewards? Having a gil fee traded to the retainer to challenge the dungeon could be a bit troublesome if that was a transfer. Same with having gil rewards. Though the gil fee could be a sink for dungeon maintenance, like with aetherytes and taxes? Maybe paying in GC seals, tomestones, leves, or some other token would be better?
Access
Where in the game world could it be added? Whether or not the entrances would be instanced I'm unsure about as well, but given the housing situation it probably should.
The dungeon itself would probably at least be public entry, and with setting a retainer one would have an in-game explanation with the option to set it private, friends, FC, and exclude blacklisted characters. Private keywords would be unnecessary if it's not instanced, and could be spread (people also may forget them). Private keys like in Bloodborne chalice dungeons would also be unnecessary as the dungeon would be player created. Player search is feasible if instanced and you don't remember (could have a note about "last visited dd/mm/yy" in the margin)
Duly noted
Something like this probably wouldn't be able to be added until at least 6.0, people are clamoring for housing space and wards regularly, would most likely use significant developer resources, and be separate from housing.
What do you think about something like player-created dungeons? If you wanted, you could make something akin to Newt Scamanders creature space in Fantastic Beast and Where to Find Them.