The design of gradient coils for a magnetic resonance imaging (MRI) system is a multiple objective optimization problem, which usually needs to deal with a couple of conflicting design objectives, such as the stored magnetic energy, power consumption, and target linear gradient distribution. These design requirements usually conflict with each other, and there is no unique optimal solution which is capable of minimizing all objectives simultaneously. Therefore, the design of gradient coils needs to be optimized reasonably with the tradeoff among different design objectives. Based on the developable property of the super-elliptical cylindrical surface and the stream function design method, the multiple objective optimization problem is analyzed by using the Pareto optimization method in this paper. The effect of proposed approach is illustrated by using the stream function method and three aforementioned coil design objectives are analyzed. The influences of the stored magnetic energy and power consumption target on linearity of gradient coil and the configuration of coils are analyzed respectively. The suitable sizes of gradient coils are discussed by analyzing the change of the stored magnetic energy. A weighted sum method is employed to produce the optimal Pareto solutions, in which the multiple objective problem reduces into a single objective function through a weighted sum of all objectives. The quantitative relationship of each design requirement is analyzed in the Pareto solution space, where Pareto optimal solutions can be intuitively found by dealing efficiently with the tradeoff among different coil properties. Numerical examples of super-elliptical gradient coil solutions are provided to demonstrate the effectiveness and versatility of the proposed method to design super-elliptical gradient coils with different coil requirements. The optimization results show that there are multiple available solutions in the convex Pareto solution space under the constraints that the linear gradient deviation is less than 5% and the magnetic stored energy and power dissipated are both no more than user-preset values. In the case that the values of summed objective functions are the same, the proposed method can intuitively see the performance of each individual target, thereby conducting to realizing the final design of gradient coils under the different design requirements. With the proposed approach, coil designers can have a reasonable overview of gradient coil design about the achievable performances of some specific properties and the competing or compatible relationships among coils properties. Therefore, a suitable design of the gradient coils for a given requirement of MRI application can be chosen reasonably.