The DU Lounge

"on" can be interpreted as follows:

we (people who dream) are the subjects of much dreaming; or
we are instruments upon which dreaming can occur.

If you replace "on" with "of," the meaning of the line is at least somewhat impoverished.

Noticing nuances like this doesn't always pay off with lesser writers, but I'm inclined to give Shakespeare the benefit of the doubt. And fwiw, from the p.o.v. of education, the skills involved in noticing and thinking about such nuances are incredibly powerful both in understanding how to express oneself clearly and also in deciphering political & other B.S.

That said, it's probably worth mentioning that no actual script of any of the plays is known to have been made available by Shakespeare himself to any publisher. Many of the plays, including Hamlet, were published only after his death, and all published versions of the plays were based on his actors' memories of the lines, who felt they should not be lost (and the plays were so popular that publishers were happy to publish them). In some cases more than one version of the same play were published with slight but potentially meaningful variations among them – part of a modern editor's job is to choose which versions to follow. In short, it's possible that "on" originated as an actor's faulty memory or even as a publisher's typo.

Another possibility is that "on" as used in the line you've quoted might have been a more common usage then than it is now. As mentioned, the first English dictionary was compiled around Shakespeare's time, and many spellings and rules of grammar were also still very much in flux (indeed, Shakespeare's own writings may have influenced how such matters evolved and were ultimately stabilized).