Right from the start, I know some people are happy with running dungeons all day as they are to cap currencies. I wouldn't be looking to remove that type of dungeon run from them. The system I'm suggesting would be an option to check off, or even a separate type of roulette. Given that a lot of time is spent repeating dungeons for currency/gear I think there could be more variation in running them.

One of the easier ways to do this would be to have variations in the types of enemies found in the dungeon and the patterns that they follow. Throw in some enemies that are really deadly that should be avoided. On rare occasions there could even be a boss style enemy that is free roaming, with the potential to encounter the party anywhere. Give him a unique loot pool and currency should it happen. Have some out of range enemies engaged in a summoning. After a certain amount of time in the dungeon they spawn a powerful demon that you can return to fight or ignore.

Randomly throw in some npc's, both helpful and in need of help. Perhaps give players an option to accept additional NPC quests inside the dungeon that opens up a new area with it's own trials and rewards. There could be lost/captured merchants that offer items that you would not find find anywhere else. There could be random minions locked away in cells that you would have to search for keys through out the dungeon to get. Drop an alchemist lab table somewhere with a random potion effect or rare dyes. Cursed tombs that when open provide rare items or materials, but inflicts a penalty on the party until the dungeon is complete.

Maybe if the party has not crossed into an area by a certain time the enemies are alerted to their presence and setup a blockade. Players could choose to descend into an otherwise locked area, with it's own set of challenges events and rewards, or fight through the blockade to return to the standard path. Perhaps when the players get to a boss room they find a hostile party of "dungeon raiders" has killed the boss and laid claim to the treasure. Make them a random composition of classes with the choice for the players fight them instead.

There are hundreds of ways and combinations of things that would make a dungeon feel interactive and give parties the decision of what to do next. It would be fun to go into a dungeon with a sense of adventure and unease about what you are getting into, instead of an autopilot mentality. Due to the added time/difficulty it would draw a smaller crowd, but I think if the rewards were balanced around that it would be a popular choice for people looking to break up the monotony of standard dungeons.