This is not a complete answer by any means, just a hint to some interesting problems where people have thought about your question.

One should bear in mind that the case of a general $X$, and drawing points at random (according to some measure), is more or less hopeless. Just take an infinite sequence of point mass (unweighted) and shape a suitable function to get divergence.
What if your sequence say escapes to infinity? As you're testing the sequence against $C_{c}$-functions, you'll get zero. You must need some recurrence of the sequence.

Moreover, you have very little to chance to prove a theorem like that, as your space probably don't carry any interesting structure to carry computations in.
Even in the basic Weyl equidistribution theorem, you know all the spectral theory and dynamical structure of your problem of rotations of the circle.

Things start to get interesting in the following scenario:

- $X$ has finite measure (although there are also some works when $X$ is infinite measure, but lets assume finite measure for the moment).
- $\mu$ is invariant under some (non-compact, say simple) group action $G$, and the points are drawn in some "nice fashion", say you are looking at orbits $G.x$.

$(2)$ is the key element here. One can then apply two powerful techniques:

- Weyl's equidistribution theorem, which relays on harmonic analysis (analysing the $G$-action on the unitary representation $G \curvearrowright L^{2}(X,\mu)$), this provides the links with automorphic representations that Marc have mentioned earlier.
- Ergodic theory (especially if $G$ is an amenable group), which would yield information such as $\mu$ a.e. point $x$ is equidistributed in the sense you've mentioned (those points $x$ are termed "generic point", due to Furstenberg). Amazing new developments (such as measure classification) enables one, sometimes, to deduce stronger results than just almost everywhere, I guess this is (partly) covered in Elon's recent survey that Marc have linked to, another nice elementary survey by Elon in this direction (of using measure classification to deduce equidistribution) - http://www.ma.huji.ac.il/~elon/Publications/Montreal.pdf.

Now I'll give a list of results (which reminds of Marc's reference) there, which will demonstrate such uses:

Zagier/Sarnak theorem that Marc have pointed to, there you're looking at $\Gamma \backslash PSL_{2}(\mathbb{R})$ where $\Gamma$ is an arithmetic lattice, the group acting the upper unipotent matrices, and you look at periodic orbits of this action with longer and longer periods (not entirely equivalent to what you've mentioned, but in the same spirit). There you can show (by analyzing Eisenstein series) equidistribution where the period tend to be large.
Recently, Einsiedler-Margulis-Venkatesh, generalized this argument in a very interesting manner, using (quantitative-)ergodic theory.
To be precise, such a theorem was already appeared in Margulis' thesis.

Furstenberg/Dani-Smilie/Strombergsson - You're facing a similiar situation to $(1)$, you are averaging over part of the horocyclic orbit - $\frac{1}{T}\int_{0}^{T}f(u_{t}.x_0)dt$, then basically unless you are trapped inside a periodic orbit (Furstenberg didn't handle those, only Dani latter), you get equidistributed (and this apply to every point, not almost everywhere statement).
This line of work was tremendously expanded by Ratner's theorems.

Linnik-Skubenko/Duke/Einsiedler-Lindenstrauss-Michel-Venkatesh - The so called Linnik problems.
The situation here is pretty much the same, $X=SL_{2}(\mathbb{Z} \backslash SL_{2}(\mathbb{R})$ or $SL_{3}(\mathbb{Z} \backslash SL_{3}(\mathbb{R})$ (notice that they are both non-compact spaces). And the group acting is the diagonal group $A<SL_{\star}(\mathbb{R})$.
It is known (at-least in the low rank case) that there are many periodic $A$ orbits which are not equidistributed, and are related to number theory (as actions of unit groups number fields).
Linnik basically bunched some of those periodics together and asked how the measure looks like if the period gets large (more specifically, as the discriminant goes to minus infinity).
He showed that this orbits gets equidistributed (his proof assumes some extra assumption, later Duke removed it via a difficult computation in the spirit of "Weyl's criterion", recently ELMV found a more natural dynamical proof which is based on measure classification).

Point counting. You would like to count the points in a "reasonable" space in a manner similar to Gauss circle problem.
In hyperbolic space, the usual comparison of area argument wouldn't work, due to the large boundary area. Then such techniques as mentioned above (a bit disguised) have shown several "equidistribution" theorems, the reference here are Duke-Sarnak/Eskin-McMullen and Lax-Phillips (in some cases of the infinite volume case).

So I'm hoping I've given you some ideas to think about...

probabilitymeasure? – François G. Dorais♦ Sep 20 '13 at 11:49