We consider a sensor selection problem for binary hypothesis testing with cost-constrained measurements. Random outputs related to a parameter vector of interest are assumed to be generated by a linear system corrupted with Gaussian noise. The aim is to decide on the state of the parameter vector based on a set of measurements collected by a limited number of sensors. The cost of each sensor measurement is determined by the number of amplitude levels that can reliably be distinguished. By imposing constraints on the total cost, and the maximum number of sensors that can be employed, a sensor selection problem is formulated in order to maximize the detection performance for binary hypothesis testing. By characterizing the form of the solution corresponding to a relaxed version of the optimization problem, a computationally efficient algorithm with near optimal performance is proposed. In addition to the case of fixed sensor measurement costs, we also consider the case where they are subject to design. In particular, the problem of allocating the total cost budget to a limited number of sensors is addressed by designing the measurement accuracy (i.e., the noise variance) of each sensor to be employed in the detection procedure. The optimal solution is obtained in closed form. Numerical examples are presented to corroborate the proposed methods.