A hysteretic subcritical period-doubling bifurcation is observed in the nonlinear strain dynamics of a magnetostrictive oscillator. The dynamic strain response of the magnetostrictive oscillator was observed with a high-resolution fiber optic interferometer. The effects of low-frequency modulation and band-limited stochastic fluctuations on such a bifurcation are investigated. Power spectral density measurements show that for an optimal value of externally injected noise the signal-to-noise ratio of a low-frequency modulation signal is enhanced by greater than 14 dB, thus indicating the first experimental observation of stochastic resonance near a bistable period-doubling bifurcation.