The Dedekind eta function shows up in three-dimensional quantum gravity: http://arxiv.org/abs/0712.0155 (Alexander Maloney, Edward Witten, Quantum Gravity Partition Functions in Three Dimensions). On page 17 a basic partition function $Z_{0,1}$ of the theory is calculated as $$Z_{0,1}(\tau)=\frac{1}{|\eta(\tau)|^2}|\bar q q|^{-(k-1/24)}|1-q|^2.$$ It also appears in the calculation of supergravity partition functions in sec.7.
The Dedekind eta function also enter in (supersymmetric) physics through mock modular forms: http://arxiv.org/abs/1208.4074 (Atish Dabholkar, Sameer Murthy, Don Zagier, Quantum Black Holes, Wall Crossing, and Mock Modular Forms).
A good review of mock modular forms is http://mathcs.emory.edu/~ono/publications-cv/pdfs/114.pdf (Ken Ono, Unearthing the visions of a master: harmonic Maass forms and number theory).
I first heared about Dedekind Eta Function in the physics context via Freeman Dyson's lovely essay "Missed opportunities": http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183533964