<p>This paper presents an equation error framework for the identification of stochastic coefficients in stochastic saddle point problems. As a key application, we demonstrate how this framework encompasses the inverse problem of stochastic parameter estimation in the nearly incompressible elasticity system, the underlying mathematical model for the elastography inverse problem of locating cancerous tumor. Beyond this, the framework is general and applies to a range of other problems. The proposed approach reformulates the nonlinear inverse problem as a stochastic quadratic optimization problem. We establish the existence and uniqueness of the solution under a quadratic regularization and analyze the impact of data perturbation. To enable numerical solution, we employ a stochastic Galerkin discretization, derive the corresponding discrete optimization problem, and provide explicit expressions for the gradient and Hessian of the objective function. To validate the framework, we apply it to a test case involving the identification of a random coefficient in a nearly incompressible elasticity system, demonstrating its capability to accurately recover both the mean and variance of an unknown stochastic Lamé coefficient.</p>