The significance of Dedekind Eta function in physics that it ensures modular invariance of certain quantities which is an important physical constraint. Dedekind eta function arises in the partition functions of string theory. The (bosonic) string partition function can be expressed as - Don't ask me what  are!! Dedekind eta function also arises in the partition function of fermionic strings, branes and supergravity theories. An enlightening viewpoint from - Urs Schreiber The basic reason is that the Dedekind eta function is one of the main examples of a modular form. These in turn are really sections of a certain canonical line bundle on the stack of elliptic curves. But since an elliptic curve over the complex numbers is just a complex torus, that means that a modular form in general (and hence the Dedekind eta in particular) is precisely something that 1. assigns something to each genus-1 closed string worldsheet;2. subject potentially a conformal anomaly.So in particular the partition function of a superstring yields a modular form (see at Witten genus for the case of the heterotic superstring) and that accounts for the bulk of the appearances of modular functions and Dedekind etas. The appearance of eta function in physics (with lots of string theory cases) is discusses in this thread - The Dedekind Eta Function in Physics