It may be easier to start with vector Green's functions for $\nabla$ and then work up to Green's functions for the Laplacian.
$G_1$, the Green's function for $\nabla$, can be easy found through the Generalized Stokes theorem.
$\oint G_1(y-x) \wedge d^{N-1}y = \int \delta(y-x) \, d^Ny = i_N$
For $N=3$, we can pick a ball for the volume integral. This relates the values of $G_1$ on the boundary of the ball to a volume integral over that ball. We can conclude that $|G_1|$ should be constant over the surface of the ball and that its direction should be radially outward. Let the radius of the ball be $R$, and we conclude that
$\oint G_1(y-x) \wedge d^{N-1}y = i_N S_N |G_1(R)|$
where $S_N$ is the surface area, and $|G_1(R)|$ signifies the magnitude of $G_1$ for any argument with magnitude $R$ (a slight abuse of notation). The result is
$i_N S_N |G_1(R)| = i_N$
so $|G_1(R)| = 1/S_N$. In 3d, this would tell us that the magnitude of the Green's function is $1/4\pi R^2$, which is absolutely true. Only a couple steps remain to build the vector Green's function. We said the direction had to be radial, so that the result is
$G_1(x-y) = \frac{1}{S_N(|x-y|)} \frac{x-y}{|x-y|}$
where $S_N(R)$ is the "surface area" of a ball with radius $R$ in $N$ dimensions.
Now, to find $G_2$, the Green's function for the Laplacian, invoke radial symmetry to find that
$G_2(x) = \int_\infty^{|x|} \frac{1}{S_N(r)} \, dr$
(Referencing to infinity here is a choice, but an incredibly convenient one for making the math work out.) Quick check: $S_3(r) = 4\pi r^2$. The result in 3d is then
$\int_\infty^{|x|} \frac{1}{4\pi r^2} \, dr = \left. -\frac{1}{4\pi r} \right|_\infty^{|x|} = -\frac{1}{4\pi |x|}$
This is indeed the Green's function for the Laplacian in 3 dimensions.