Strain monitoring is of great significance to identify the failure of key mechanical components and ensure the good operation of mechanical equipment. In terms of the common issue of the low sensitivity of fiber Bragg grating (FBG) strain sensor in strain measurement on the mechanical structure surface, this paper describes a flexible hinge strain sensor with the FBG as the sensitive element. A theoretical analysis of the sensitivity of the sensor is performed, a copper alloy material with a large elastic modulus is selected, the simulation analysis and optimization design for the structural parameters of the sensor are carried out by using ANSYS and Solidworks software, then the object of the sensor is developed according to the simulation results, and finally, the performance of the sensor is tested. The results show that the maximum deformation the sensor can measure is 0.5299 µε, the sensitivity of the sensor is ∼1.8870 pm/µε, which is approximately twice that of the bare FBG sensor, the linearity is greater than 99%, and the relative standard deviation of the repeatability is 0.9%. The results of the study provide a reference for developing the optical fiber strain sensor and further improving its sensitivity.