In this paper we propose a finite element method for solving elliptic equations with observational Dirichlet boundary data which may subject to random noises.The method is based on the weak formulation of Lagrangian multiplier and requires balanced oversampling of the measurements of the boundary data to control the random noises.We show the convergence of the random finite element error in expectation and,when the noise is subGaussian,in the Orlicz ψ2-norm which implies the probability that the finite element error estimates are violated decays exponentially.Numerical examples are included.