The paper that appears to have introduced the _term_ "softmax" is [Training Stochastic Model Recognition Algorithms as Networks can Lead to Maximum Mutual Information Estimation of Parameters][1] (1989, NIPS) by John S. Bridle.

As a side note, the [softmax function][2] (with base $b = e^{-\beta}$)


$$\sigma (\mathbf {z} )_{i}={\frac {e^{-\beta z_{i}}}{\sum _{j=1}^{K}e^{-\beta z_{j}}}}{\text{ for }}i=1,\dotsc ,K {\text{ and }}\mathbf {z} =(z_{1},\dotsc ,z_{K})\in \mathbb {R} ^{K}$$

is very similar to the [Boltzmann (or Gibbs) distribution][3]

$$
p_i=\frac{e^{- {\varepsilon}_i / k T}}{\sum_{j=1}^{M}{e^{- {\varepsilon}_j / k T}}}
$$

which was formulated by Ludwig Boltzmann in 1868, so the idea and formulation of the softmax function is quite old.

 [1]: https://papers.nips.cc/paper/195-training-stochastic-model-recognition-algorithms-as-networks-can-lead-to-maximum-mutual-information-estimation-of-parameters.pdf
 [2]: https://en.wikipedia.org/wiki/Softmax_function
 [3]: https://en.wikipedia.org/wiki/Boltzmann_distribution