The Somnium Scipionis, a text sometimes extracted from Cicero's Republic on its own, has been called the "first Western dream vision." There's a little bit of modern categorization tapdancing going on there, as is evident in the text itself. But it's relevant here for a couple of reasons. First, it forms the basis of Western dream theorizing going forward, most prominently in the Middle Ages (where the nature of dreams, and whether they could ever be revelatory, was a very important topic for theologians). Second, Cicero shows a pretty keen awareness of the craziness of dreamland.

He recounts a story of Scipio Africanus the Younger, who had spent an evening discussing his grandfather (that would be, the Elder version) with an acquaintance:

Then, when we had parted to take our rest a sleep much deeper than was usual fell upon me, for I was very weary from my journey and had stayed awake until very late. And then—(I suppose it was a result of what we had been talking about; for it happens often that the things that we have been thinking and speaking of bring about something in our sleep. So Ennius relates in his dream about Homer, of whom in hours of wakefulness he used so often to think and speak)—Africanus stood there before me, in figure familiar to me from his bust rather than from life.

(Fragments of Ennius' dream of Homer, part of the prologue to his Annals, are discussed in this article.)

I don't mean to oversell the "reality" of Scipio Jr. having this dream or telling Cicero about it. Rather, we can look at the fact that Cicero included this story that would have been plausible to his readers, for information on the sorts of ideas that circulated about dreams in the Roman Empire.

By the late Middle Ages, the dream vision is a stock literary trope for structuring a whole poem. Classical allusions (fiction, mythology) fly everywhere, including references to multiple versions of the same story (which critics generally argue either makes the poem more dream-like, or has some Bigger Literary Significance In That Poem). Obviously something like Chaucer's House of Fame is wildly removed from any recoverable act of slumber, but certainly the idea is well entrenched.

One interesting point about the contentious relationship between actual dreams and religious revelation is that the long tradition of medieval women visions almost never have their visions while asleep. In ecstasy or in a fever haze, sure; but only in the rarest cases asleep. Even the visions of heaven and hell so popular in the early Middle Ages tended to have protagonists undergoing more of a near death experience than a good solid nap.

Dreams could have meaning in the sense of conveying moral lessons, but divine revelation was a very, very touchy subject.