Hysteresis is the parameter which is used to make neighbor cell look worse than the current measured value. It helps to UE to make sure that neighbor cell is strong enough and if network triggers handover after receiving the measurement report from the UE, handover to target cell will be successful.
Offset is the parameter that make serving cell look better than the actual measurement of it compare to neighbor cell.
Threshold is used when measurement of both the cells are not compared and while taking decision to trigger measurement report both or one cell measured independently. For example : Event A1, Event A2 .