The cytoskeleton is responsible for determining the mechanical response of living cells, and its remodeling emerges as a key biomarker in carcinogenesis and increased cell malignancy. Its architecture deeply relies on the assembly of filamentous actin with crosslinker proteins, establishing such networks as a crucial driver of cell elasticity and structural function. Due to the dynamic nature and inherent microstructure uncertainties of these networks, deterministic models often fall short in accurately capturing their complex behavior, highlighting the need for stochastic approaches to obtain a more realistic representation of the cytoskeletal mechanics. In this work, a continuum model of F-actin networks with compliant crosslinkers is employed to characterize the mechanical response of the actin cortex. Network uncertainties are introduced using Polynomial Chaos Expansions. These provided the foundation for two distinct approaches: a non-intrusive approach for constructing surrogate models and an intrusive approach for developing a Stochastic Finite Element Analysis framework. This methodology not only validated the effectiveness of stochastic modeling for capturing the variability in cytoskeletal mechanics, but also offered deeper insights into the influence of model parameters, highlighting which are most critical to treat as stochastic. This framework lays the groundwork for future developments, including the incorporation of crosslinker dynamics.