Riemann used this notation in his zeta function paper.

Riemann uses the notation $s=\frac12+it$ for $s$ on the critical line, but I cannot find any appearance of $\sigma$ in his paper. On the contrary, the notation $a+bi$ appears often there.

However, the following sentence occurs in the first paragraph of Ivic's book:

Riemann wrote $s=\sigma+it$ ($\sigma,t$ real) for the complex variable $s$, and this tradition still persists, although some authors prefer the more logical notation $s=\sigma+i\tau$.

I cannot remember where, but I vaguely recall reading that the tradition was initiated by Landau's book.