In this paper, an energy-dissipating steel-hinge connection in a prefabricated joint was proposed, in which hole-opened weakened steel-plate dampers were its key parts. The axial reciprocating loading tests were conducted on three different hole-opened weakened dampers to investigate the failure mode, the hysteretic performance, skeleton curve, bearing capacity as well as the ductility. Then the effects of the weakened length and width of the hole, the width-thickness ratio, and the thickness direction gap on the hysteretic performance of the steel-plate damper were discussed. In addition, a simplified mechanical model of the weakened steel-plate damper was established to derive the hysteretic constitutive model which was verified by the experimental results. The results show that the weakened steel plate of the damper cracks or breaks at the hole-opened part, avoiding out-of-plane buckling and achieving the controllability of plastic energy dissipation and failure mode. The hysteresis curve of the dampers is full, the bearing capacity is higher than 297.31 kN, and the displacement ductility coefficient is greater than 4.5, showing good energy dissipation capacity, bearing capacity and ductility performance. Compared with diamond-shaped openings, the vertical slot-opening damper has better overall mechanical properties. It is also recommended that the weakened length a/L of the hole is from 0.25 to 0.55, the weakened width b/B is from 0.2 to 0.5, the width-to-thickness ratio is from 12.50 to 15.63, and the gap in the thickness direction cannot exceed 2 mm. The proposed hysteretic constitutive model can accurately simulate the hysteresis performance of the hole-opened weakened steel-plate damper.