I've spent the last half hour browsing Stillwell's translation of Poincaré's Analysis Situs and Dieudonné's History of Algebraic and Differential Topology, and I haven't found the source of this notation. The "1" in the subscript would lead one to believe this notation was introduced at the same time as $\pi_n(X,x)$ for the higher homotopy groups as well.
In my search, I have learned that the name "Fundamental group" comes straight from Poincaré, and from the "Fundamental region" of a group action. (And, incidentally, that "torsion" in algebra comes from the homology of non-orientable surfaces, which are twisted in on themselves. A bit of a tangent, but I'm still marveling.)