In some cases, you can recover L_2 information in the limit by considering random walks, or said differently, basically by counting paths. When you say "discrete space with L_1-like behavior," I imagine a big grid of points (something like Z^d) that are connected to their nearest neighbors by edges, with the distance between two points being defined in the natural graph-theoretic way as the length of the shortest path between them. If that's the case, you can consider a uniform random walk along the edges of the graph, and then successively refine the grid. This is important: if you refine the grid, say, by a factor of two (so each step of the random walk becomes half as big), then you have to refine the walk by taking steps four times as fast. (In general, when you refine by a factor F, you have to take steps F^2 times as fast.) As you keep refining, the probability of the walk ending up near a particular point at a fixed time in the future may stabilize to something that looks like a function of L_2 distance from the starting point.
If this construction works in your case, then you can think of the logarithm of the probability of ending up in a small neighborhood as being roughly proportional to the volume of the neighborhood and the square of its L_2 distance from the starting point, but except in very special cases, the approximation probably won't be good for all pairs of points, especially if they aren't very close to one another. Note that computing such probabilities basically amounts to counting paths of particular lengths between points, as opposed to just finding the length of the shortest path (which would roughly corresponding to the L_1 distance).
How much that computation will actually resemble an L_2-like distance depends on many things. The construction will work basically as stated when you successively refine a grid like Z^d because of the central limit theorem. If your "discrete space" isn't very similar to that, then the construction might not work at all, or it might give you something that isn't quite right but is "close enough for government work," so to speak. You'll have to be the judge of that.