This paper presents a theoretical foundation for designing the power and bandwidth performance of an electromagnetic nonlinear vibration energy harvester with a hardening resonator. To this end, the steady-state solution derived via first-order approximation is investigated to establish a graphical approach, which gives a clear perspective on how the resonance peak point on the frequency–displacement curve is determined in terms of the excitation amplitude, nonlinear restoring force function, and other design parameters. Then, a ν-power bandwidth, a generalized version of the half-power bandwidth, is newly introduced and its approximate analytical formulation is derived. Based on the findings of the parameter study on the ν-power bandwidth, a design scheme is proposed, which begins with specifying the location of the resonance peak point that guarantees the existence of the high-energy branch under possible variations of the excitation, followed by determining the linear natural frequency that makes the ν-power bandwidth as large as reasonably possible. Design examples, assuming a specific class of hardening nonlinearity of the restoring force, are shown to demonstrate how the proposed design scheme yields vibration energy harvesters with maximized power bandwidth performance while incorporating given design requirements.