We consider the numerical solution of an inverse problem for the Laplace equation where the Robin coefficient is to be recovered from a partial boundary measurement. We first formulate the problem by integral equations on the boundary. By introducing a new variable, the key integral equation involved becomes linear, and the ill-posedness of the inverse problem is clearly revealed. On the basis of this linearity, we then design linear least-squares-based methods for reconstruction of the Robin coefficient. A proper form of regularization is chosen, and both direct (non-iterative) and iterative methods are investigated. Numerical examples are presented to show the effectiveness of these methods in providing excellent estimates for the Robin coefficient from data with and without noise.