Novel convex measurement cost minimization problems are proposed based on various estimation accuracy constraints for a linear system subject to additive Gaussian noise. Closed form solutions are obtained in the case of an invertible system matrix. In addition, the effects of system matrix uncertainty are studied both from a generic perspective and by employing a specific uncertainty model. The results are extended to the Bayesian estimation framework by treating the unknown parameters as Gaussian distributed random variables. Numerical examples are presented to discuss the theoretical results in detail.