We show numerically that under certain conditions noise-induced soliton self-mode conversion dominates over soliton self-frequency shift for a soliton in a high order spatial mode of a multimode optical fiber. The input soliton has to be group index matched to a lower order mode for a frequency separation for which the Raman gain is non-negligible, and this condition determines the wavelength of the pulse growing from noise. The phenomenon has no known analogs in single-mode or graded-index fibers. The results demonstrate that it is possible for a noise-induced physical process to dominate over a seeded one even for noise levels at the fundamental limit.