I think extra story content to go along with the optional dungeon is nice, mostly because it provides extra motivation to see it through. For example, I once played a game where a few of your party members end up dying in the main story at various points. In post game there is an optional dungeon that sort of is like an afterlife and you meet some of those party members there and they team up with you briefly and also sort of come to terms with how their life played out. FF II had a section like this as well, its mostly just something to provide some context to the extra content.
For some, achievements like on console or steam are enough to see something through. The items like Eschaton mentioned were often done in older games because no such thing existed, It was your way to show your buddy, "hey I beat that tough boss." Achievements let "the world" see that and that is something that matters for some players.
For some challenge is enough, they like having to push themselves further in the game, test their skills even more.
There's also something like the Star Ocean games and games by their developer are known for, the game truly opens up its systems in post game; you need to know the crafting system inside and out to make the best gear, you end up having even a couple hundred levels left to get. This is ultimately a way to enjoy the game and what it offers after you beat it, make it feel like you are starting out at the bottom again. It sort of becomes a time sink though especially if there's no real story content to go with it like the main game and if you don't enjoy playing the game that much, the the player might just drop it eventually.
Essentially an optional area is optional, you can try to motivate the player to complete it or see it but some might just not do that.