The move from betti numbers to homology groups.
Although this might not fit super tightly with the usual modern examples of "categorification" (in the way that say a monoidal category is a categorification of a monoid), it is probably the first and most important example of a concept being categorified, allowing for notions such as functoriality, naturality, etc. to flourish. [No way for a continuous map between spaces to induce a map between betti numbers! The old days before functoriality!].