Continuum manipulators, with their characteristics of flexibility and dexterity, have gained significant interest in various applications across industries such as inspection, manufacturing, space exploration, and medical surgery. However, because of their inherent compliance, handling payloads may prove challenging due to shape distortion and deflection. This demonstrates the need to optimize the manipulator’s stiffness. The primary objective of this work was to show the merits of sensitivity analysis in the design of flexible surgical manipulators. Such analysis can guide important design decisions and enable the more efficient use of available resources, contributing to designing more effective prototypes. A new sensitivity analysis framework based on a multi-model and a multi-method approach was proposed to achieve this. This framework was then demonstrated by studying a tendon-driven rolling contact joint hyper-redundant manipulator for transoral laser microsurgery. In this analysis, the effects of independent design parameters on the stiffness of the manipulator were examined. Then, scaled-up 3D-printed prototypes were used to validate the accuracy of the stiffness model experimentally, which enabled us to assess the outcome of the sensitivity analysis framework. The results demonstrated that only two out of five design parameters for the considered manipulator significantly impacted the device’s performance. This information could enable the designer to efficiently allocate resources toward correctly setting these two most important parameters to achieve the desired system. Overall, the proposed analysis framework is a general tool that can be applied to any design architecture, helping to develop optimal manipulators for various applications.