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I am asking about the reason for the use of this letter, if known, as well as the initial occasion of its use. Ideas that have been suggested concerning the former include:
As for the latter, I'm guessing it originates somewhere between Riemann and Einstein.
You are guessing correctly. Riemann did not use $g$ for the metric tensor, he writes things like $ds^2$ or $\sum dx^2$ instead, see his 1854 lecture "On the Hypotheses which lie at the Bases of Geometry" (1854).
Originally, "g" was for gravity. Einstein and Marcel Grossman, his mathematician friend who introduced him to tensor calculus and collaborated with him on prototypes of general relativity, adopted it after realizing some time between 1909 and 1913 that the metric tensor depended on the distribution of gravitating matter in a region of spacetime. After the success of Einstein's final version of the theory the notation was widely adopted by physicists and spread into tensor calculus and differential geometry as well. See Foster and Nightingale's Short Course in General Relativity (p.112).
