The next awaited breakthrough in proton therapy is the inclusion of the tools for online monitoring of beam range into clinical practice. Full, three-dimensional information on the deposited dose distribution can be obtained by means of prompt gamma imaging using Compton cameras. Large gamma detection efficiency and high-rate capacity can be achieved using detectors of high granularity made of a heavy scintillator. One of the possible design options is a stack of scintillating fibers. As the overall performance of such a camera depends on the position, time and energy resolution of the fibers, we investigate those properties both experimentally in measurements with a test bench as well as via Monte Carlo simulations. The obtained results point at LYSO:Ce as the best candidate for a sensitive material of a Compton camera of the discussed type.