[en] A 56 MHz superconducting RF cavity was designed for a luminosity upgrade of the Relativistic Heavy Ion Collider (RHIC), including requirements for Higher Order Mode (HOM) damping. In this paper, we describe our optimization of the damper's performance, and modifications made to its original design. We also show the effects of the damper geometry on the cavity's HOM impedance. To reduce the likelihood of magnetic breakdown, we lowered the magnetic field enhancement at the ports to a value less than the highest field in the cavity. We simulated all monopole and dipole HOMs up to 1GHz with their frequencies, mode configurations, R/Qs, and shunt impedances, verifying that all modes are well-damped with the optimized design and configuration. The 56 MHz superconducting RF cavity is a quarterwave resonator designed to have a gap voltage of 2.5 MV. Our plans are to place this beam-driven resonator at a common section of RHIC to provide a storage RF potential for both rings. The large bucket of the cavity will reduce spill due to Intra-Beam Scattering (IBS), and thus increase the luminosity for the detectors. It is very important to damp all the cavity's Higher Order Modes (HOMs) to avoid beam instabilities. The design chosen for the HOM damper is a magnetically coupled loop located at the rear end of the cavity. The loop and its port geometry must be optimized to assure sufficient damping, avoid a large enhancement of the local magnetic field. A high-pass filter is included in the circuit to reduce the power extraction from the fundamental mode. The number of HOM dampers used and their configuration also are important factors for the damping and cooling system. A small loop area will couple out less power from the cavity's fundamental mode, thus reducing the voltage and power dissipation in the damper's filter circuit; however, it might not be sufficient for HOM damping. This problem is resolved by increasing the number of the HOM dampers and carefully choosing their location. Details of the high-pass filter will be discussed in another paper.