Stochastic homogenization (with multiple fine scales) is studied for a class of nonlinear monotone eigenvalue problems. More specifically, we are interested in the asymptotic behaviour of a sequence of realizations of the form $$-div\left(a\left(T_1\left(\frac{x}{{\epsilon }_1}\right){\omega }_1,T_2\left(\frac{x}{{\epsilon }_2}\right){\omega }_2,\nabla u_{\epsilon }^{\omega }\right)\right)={\lambda }_{\epsilon }^{\omega }𝒞\left(u_{\epsilon }^{\omega }\right).$$ It is shown, under certain structure assumptions on the random map $a({\omega }_1,{\omega }_2,\xi )$, that the sequence $\{{\lambda }_{\epsilon }^{\omega ,k},u_{\epsilon }^{\omega ,k}\}$ of $k$th eigenpairs converges to the $k$th eigenpair $\{{\lambda }^k,u^k\}$ of the homogenized eigenvalue problem $$-\mathrm{div}\left(b\left(\nabla u\right)\right)=\lambda \overline{𝒞}\left(u\right).$$ For the case of $p$-Laplacian type maps we characterize $b$ explicitly.