After some searching I found some relevant references:
H. Rademacher and A. Whiteman, “Theorems on Dedekind sums,” Amer. J. Math. 63 (1941), 377–407.
Arias-de-Reyna, J. Riemann's fragment on limit values of elliptic modular functions.
Ramanujan J. 8 (2004), no. 1, 57--123.
Dedekind's paper is titled "Erlauterungen zu den vorstehenden Fragmenten" and was published published in Riemann's collected works. In this paper Dedekind attempts to elucidate Riemann's “Fragmente uber die Grenzfalle der Elliptischen Modulfunctionen,”
It looks like Dedekind's introduction of the eta function was inspired by these fragmentary notes of Riemann. It would take me some time to parse the relevant issues (e.g. to what extent did Riemann know the Dedekind eta function, how might things have seemed to Dedekind, etc.) and I will not do so at the moment, but am happy to have found these for future reference.
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