I think the paper "Abstract Hilbert Schemes" by Artin and Zhang might be useful. This may be one of the other papers Peter had in mind in his answer.

The basic idea as far as I understand it seems to be the following; things work nicely if your ring is strongly noetherian! This condition means that whenever you tensor your ring with a commutative noetherian ring your ring is still noetherian. There are examples of rings which are noetherian but not strongly noetherian (the naive blow-ups of Rogalski), but the quantum plane is certainly strongly noetherian - it is an Ore extension and when you tensor with a commutative ring you can express the result as another Ore extension. Then you can use the version of Hilbert's Theorem for Ore extensions, Theorem 2.6 in Goodearl and Warfield's "Introduction to Noncommutative Noetherian rings".

Back to Artin and Zhang's paper, quoting from the abstract, they show that "For the category of the graded modules over a strongly Noetherian graded ring, the Hilbert functor of graded modules with a fixed Hilbert series is represented by a commutative projective scheme."

There is a recent paper of Nevins and Sierra which I think looks at what can be said for non-strongly noetherian rings, http://arxiv.org/abs/1009.2061. I think that they may work with just point modules though, these being the graded modules with Hilbert series that of a polynomial ring in one variable.

I hope this is the kind of thing you were looking for. Also, I'm sure that someone more expert than myself would be able to explain in more detail what Artin and Zhang do.