I wonder if there is a way to extend the finite difference discretization of the Laplacian on a uniform grid to a nonuniform grid.
More specifically, I am not sure that the finite differences approach is the right approach for my problem, where I have a plane discretized using rectangles of different sizes, that meet at T junctions (see for example this illustration).
Is finite differences a good approach for computing the Laplacian on this grid, or is there a better one?
Not only is your grid non-uniform, it also has T-junctions. You are right that finite differences do not sound like a good approach in this setting.
I would take a look at the computational fluid dynamics literature: it is very common there to discretize space into octrees similar to your mesh, and discretizations of the Laplacian have been derived and thoroughly evaluated in this setting, since it is needed for the pressure projection step of the fluid simulation. This paper (which is surely not canonical, but is the first paper I found when doing a quick search) suggests using trilinear elements (bilinear in 2D) centered on the non-T-junction nodes: http://epubs.siam.org/doi/pdf/10.1137/090747774
With a standard finite element method for the Laplacian, you could devise a mesh as follows, with triangles at the T-junctions and quadrilaterals anywhere else:
A useful reference may be :
Simplify Laplace equation in rectangle geometry (2nd answer is FEM) .
