(Just to make it clear, I'm talking about dreams as in REM sleep, not lifelong wishes which are already in.)
In the very early (unreleased) versions of DF, there used to be a wizard that would show up to the fortress to request a specific dwarf to go on a journey and if you refused, someone in your fortress would have a "terrifying dream about an army of the dead" shortly before your fortress getting invaded by, well, said army of the dead. Of course this was all very scripted and this part of the code is now inactive, but I believe this predictable system could be expanded into a procedurally generated one.
Basically, sentient creatures (or any kind of creature really? I believe animals can dream, though I'm not sure which ones) should be able to experience dreams when they are unconscious. You'd be able to see it in lieu of their current "thought of the moment" in the thought screen. The description of the dream would look very much like item descriptions depicting common historical events, except the depictions here would be somewhat random (ideally they'd make sort of sense in the weird way dreams do, that is, not at all outside of a context) For instance, "Urist Uristurist is dreaming of the theft of the forgotten beast Edros Urbrask's left foot, fourth toe by the kobold Thiskklikumis in the Cave of Eternity. Urist is making a plaintive gesture. The kobold is secreting freakish grey muck. The kobold is wearing a giant cave spider bone hat. On the hat is an engraving of Urist's mother. It menaces with spikes of elf leather. On the hat is a coating of the goblin Sokol Sklmuis's vomit." And so on. The dreams would usually be about recent events and depicting objects and people known to the dreamer (whether by personal acquaintance, worship, recent killings, etc.) and could betray the thoughts, needs and current concerns of the dwarf. A dwarf could feel a wide array of emotions after any particular dream, from relief after a nightmare to fondness after a dream of romance.
Now the reason such a system could be worthwhile on top of fleshing out already complex personalities is that sometimes, not very often, some dwarves would be struck by premonitory dreams. These dreams could alert the player of an imminent danger (from the arrival of a titan to an impeding siege, or the so-called "army of the dead" if we're talking about a necro siege, or even just a deadly evil cloud) and trigger an announcement so the player takes it seriously. I guess there could be a system where some dwarves are naturally more sensible to these dreams than others (and this could be derived from their personality and values) and have them more often. It could also be related to religious aspects, for instance a dwarf that worships a god of death could have a premonition of a terrifying army of the dead (again) laying siege to the fortress, a god of war could warn about goblins, and so on. There could be an increased incidence of such dreams during full moon. Some forgotten beasts could also induce dreams in dwarves who worship them, driving them mad from behind their sealed drawbridges, thus giving the player an incentive to go and fight them. In any case, the possibilities are numerous and I think I'm only scratching the surface.
The best thing is that the framework shouldn't be that hard, it's basically just changing the thought screen to a procedurally generated engraving description from time to time, and then it's easy to go from there to actual, scary business.
Anyway, I did a (admittedly quick) search for similar suggestions but only found some related to lifelong goals, not actual dreams, due to the multiple meanings of the word, so sorry if I missed an existing thread.