I think that a good dungeon tells a story. I liked how Praetorium did this when I was starting up. You've got a nice build-up towards the final boss fight with some recognizable villains. The vehicular section was fun at the time, especially for the FF6 nostalgia. It also needs atmosphere. Haukke, Gubal, and Antitower are good examples of using music, interesting visual elements, and even hidden lore snippets in book piles to add to the experience.
It's harder to create an interesting boss fight under the constraints of a dungeon, because you only get to fit a couple of mechanics into an encounter. Demon Wall (AK) was probably the most memorable dungeon fight for ARR players, given its difficulty at the time and its reputation from prior entries in the FF series. While it's mechanically quite simple, I think part of what made it interesting (and what made the later variations less so) was the fact that you could fall off and be removed from the fight. How would I improve on it? Bring it back in the next Hildebrand questline in a spoof dungeon. Might as well put the Trust system to good use. If we can't see other players get flung off into bottomless pits, at least we can have some NPCs do it for us.
I think the downside of bringing in big ticket fights into dungeons is that you end up devaluing the encounter. Lahabrea and Igeyorhm is a fight that could have had a lot more potential if it wasn't stuffed into the tail end of the Aetherochemical Research Facility, for example. That's the sort of thing that deserves to be its own trial or raid encounter. But maybe Pandaemonium will fix this and produce some high quality Ascian showdowns.
My favorite dungeon experience is the original Wanderer's Palace. When you go back to ARR dungeons, it's amazing to see how even having simple branches in dungeon layouts really opens it up and makes the space feel less like a corridor. I don't care if I never pull a particular pack of mobs. That's fine. It likely takes all of five minutes to create a second branch path with an equivalent set of mobs. And the giant Tonberry Stalkers hunting you down created a sense of urgency as you went around looking for oil to unlock the route forward.
I would love to see this mechanic implemented in the next Deep Dungeon. I've always loved Persona 3's Tartarus climb, with the Reaper showing up when you dally too long in any particular floor. Can you imagine it? You're on B87 in the gloomy basements of Wanderer's Palace, when you get a floor with a darkness debuff that blacks out everything except for a small circle of lamplight around you. You spend a few minutes looking around for oil to open the exit door. A lamp quietly lights behind you, glinting off a sharpened knife. It's a giant Tonberry stalker! Frantic pursuit ensues. It would be the best.
The dungeon that I have the most hopes for next expansion is Phantom Train. FF7 had a fantastic section where you had to stop a runaway train while on a timer. I hope we see some sort of homage to that. I also hope that Pandaemonium ends up being a properly explorable dungeon-like zone, complete with a final save point before the end raid boss. If you're going to pull a Final Fantasy end game, you have to do it right.