One reasonable thing to note is that defining $$f'(x, y)=f(x+1, y+1)+f(x-1y)+f(x-1, y-1)-f(x+1y)-f(x, y-1)-f(x-1y-1)-f(x, y+1)$$ we have that $f'$ is also harmonic in this sense. So we have that arbitrary "derivatives" of harmonic functions are harmonic, though this is much more trivial than in the smooth case.
I don't see how to get the smooth case from this sort of remark, but I do think that this captures the notion of the derivative of a harmonic function to some extent, in that it measures some sort of rotation and scaling (as you point out in your answer to the math.SE question).
