Looking at it, I would say it is the energy caused by turning on positional correlations.
Let's say that there aren't spatial correlations. Then the no correlation energy is $E_{n.c.} = \frac{1}{2} \int n V(r) dV$, where the quantities are defined as in your question.
Now lets suppose we put an electron on the origin and turn on correlations. The density of the other electrons will no longer be uniform because of attraction and repulsion with the electron at the origin. But by symmetry, we expect the electron density function to depend only on the distance from this electron we have put in. One can express this radial dependence with a function $g(r)$ which has the following properties: $g(r) \to 1$ as $r \to \infty$, and $g(r)$ is proportional to the electron density. Another way of saying this is that $g(r)$ is the electron density perturbed by correlations, normalized by the unperturbed density (since the perturbed and unperturbed densities should be equal at infinity where no one knows about the electron at the origin). With this in place, the correlated electron density is $\rho_c(r) = n g(r)$. The energy of this electron configuration is $E_c = \frac{1}{2} \int \rho_c(r) V(r) dV = \frac{1}{2} \int n g(r) V(r) dV$.
Now lets find the energy from turning on positional correlation. After turning on correlations, there is an energy $E_c$, but before turning on correlations, there was an energy $E_{n.c.}$. Thus there was an energy difference $E_c - E_{n.c.} =\frac{1}{2} \int n g(r) V(r) dV - \frac{1}{2} \int n V(r) dV = \frac{n}{2} \int (g(r) -1)V(r)dV $. This is just a guess though. Tell me if it makes sense.
The $g(\mathbf r_1, \mathbf r_2)$ is defined as
$$g(\mathbf{r}) = \frac{\rho^{(2)}(\mathbf{r}_1,\mathbf{r}_2)}{\rho^{(1)}(\mathbf{r}_1) \rho^{(1)}(\mathbf{r}_2)}$$
where
$$\rho^{(n)} (\mathbf r_1, \dots, \mathbf r_n) = \frac{N!}{(N-n)!} \frac 1 Z \int e^{-\beta V} d \mathbf r^{(N-n)}$$
If the system is homogeneous,
$$\rho^{(1)} (\mathbf r) = \rho \ \ \ \ \text{(bulk density)}$$
so that
$$g(\mathbf{r}) = \frac{\rho^{(2)}(\mathbf{r}_1,\mathbf{r}_2)}{\rho^2}$$
and if the system is also isotropic,
$$g(\mathbf r_1, \mathbf r_2) = g(\mid \mathbf r_1-\mathbf r_2\mid) = g(r)$$
So we can interpret $g(r)$ as the probability to find a particle in a volume $d \mathbf r$ around a chosen particle, and $g(r) \rho d \mathbf r$ as the average number of particles in the volume $d \mathbf r$.
Now, it can be shown that
$$\rho g(\mathbf r) = \frac 1 N \langle \sum_{i\neq j} \delta (\mathbf r + \mathbf r_i - \mathbf r_j) \rangle $$
The structure factor is defined as
$$S(\mathbf k) = \frac 1 N \langle \sum_{i,j} e^{-i \mathbf k \cdot (\mathbf r_i - \mathbf r_j)} \rangle$$
so that you have
$$S(\mathbf k) = 1+\frac 1 N \langle \sum_{i\neq j} \int d \mathbf r e^{-i \mathbf k \cdot \mathbf r} \delta (\mathbf r + \mathbf r_i - \mathbf r_j) \rangle = 1+\rho \int d \mathbf r e^{-i \mathbf k \cdot \mathbf r} g(r) = 1 + \rho \tilde g(\mathbf k)$$
where $\tilde {(.)}$ is the Fourier transform.
$h$ is defined as
$$h(r)=g(r)-1$$
So that
$$S(\mathbf k) = 1 + \rho \tilde h(\mathbf k) + (2 \pi)^3 \rho \delta(\mathbf k)$$
which for $\mathbf k \neq \mathbf 0$ becomes
$$S(\mathbf k) = 1 + \rho \tilde h(\mathbf k)$$
For a more complete exposition, I would suggest Theory of Simple Liquids by Hansen and McDonald.
The confusion arises from the fact that, while $h(r)$ is a dimensionless quantity, its Fourier transform $\tilde h(\mathbf k)$ is not: it has the dimension of a volume.
Best Answer
The function $g(\mathbf{r})$, or more properly $h(\mathbf{r})=g(\mathbf{r})-1$ which tends to zero at large separation, is usually termed the pair correlation function. Only when the system is isotropic, such as a liquid, do we usually refer to $g(r)$ as the radial distribution function, since it depends only on the magnitude, not the direction, of the separation vector: $r=|\mathbf{r}|$.
You can find the full definitions in books such as Chaikin and Lubensky, Principles of condensed matter physics. I'll just give a short summary here. The general definition of $g$ takes into account its dependence on both of the positions under consideration. If the instantaneous number density of $N$ atoms is written as a sum of 3D Dirac delta functions of the atomic positions $$ n(\mathbf{r}) = \sum_{i=1}^N \delta(\mathbf{r}-\mathbf{r}_i) $$ then the pair correlation function $g(\mathbf{r}_1,\mathbf{r}_2)$ is defined such that $$ \langle n(\mathbf{r}_1) \rangle \langle n(\mathbf{r}_2) \rangle \, g(\mathbf{r}_1,\mathbf{r}_2) = \langle n(\mathbf{r}_1) n(\mathbf{r}_2) \rangle - \langle n(\mathbf{r}_1)\rangle \delta(\mathbf{r}_1-\mathbf{r}_2) $$ This has the interpretation that $g(\mathbf{r}_1,\mathbf{r}_2)$ is the probability density (per unit volume), given an atom at $\mathbf{r}_1$ of finding a different atom at $\mathbf{r}_2$, normalized by the number density in both places. Going from this to the radial distribution function involves two stages. Firstly, when the system is homogeneous and translationally invariant, one can express $g$ as a function of the difference in position vectors, $g(\mathbf{r}_1-\mathbf{r}_2)\equiv g(\mathbf{r})$. In this case, $\langle n(\mathbf{r})\rangle=\langle n\rangle$, independent of $\mathbf{r}$. We may integrate over one of the coordinates, and the result may be expressed $$ \langle n \rangle \, g(\mathbf{r}) = \left\langle \sum_{i\neq 1}^N \delta(\mathbf{r}-\mathbf{r}_i+\mathbf{r}_1) \right\rangle $$ The probabilistic interpretation is even clearer here: both sides represent the density of (other) atoms per unit volume, at a position $\mathbf{r}$ relative to atom $1$, given that the latter is at $\mathbf{r}_1$.
Secondly, when the system is, additionally, isotropic, one can express $g$ as a function of the separation distance $g(r)$, where $r=|\mathbf{r}|$, and this is what we call the radial distribution function.
The radial distribution function does not give complete information about pair correlations in crystals because they are neither homogeneous nor isotropic. Of course, one is still free to define translationally and rotationally averaged versions of the full pair correlation function of a crystal, which give some useful information about the typical surroundings of the atoms.