Using optical measurements, we demonstrate that the rotation of micron-scale graphene nanoplatelets levitated in a quadrupole ion trap in high vacuum can be frequency-locked to an applied radiofrequency electric field E-rf. Over time, frequency-locking stabilizes the nanoplatelet so that its axis of rotation is normal to the nanoplatelet and perpendicular to E-rf. We observe that residual slow dynamics of the direction of the axis of rotation in the plane normal to E-rf is determined by an applied magnetic field. We present a simple model that accurately describes our observations. From our data and model, we can infer both a diamagnetic polarizability and a magnetic moment proportional to the frequency of rotation, which we compare to theoretical values. Our results establish that trapping technologies have applications for materials measurements at the nanoscale.