As Jim Humphreys has suggested in the comments, practically all of Gian-Carlo Rota's career could be described as breathing new life into unjustly neglected subjects: Möbius functions of posets, invariant theory, lattice theory, etc. For the purposes of MO, let me single out the umbral calculus as a specific subject that languished and was revived by Rota. For anyone who is skeptical of the power of umbral calculus, I recommend Gessel's paper on applications of the classical umbral calculus. Gessel writes:
When I first encountered umbral notation it seemed to me that this was all there was to it; it was simply a notation for dealing with exponential generating functions, or to put it bluntly, it was a method for avoiding the use of exponential generating functions when they really ought to be used. The point of this paper is that my first impression was wrong: none of the results proved here (with the exception of Theorem 7.1, and perhaps a few other results in section 7) can be easily proved by straightforward manipulation of exponential generating functions.
