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Summary This paper studies the use of a new nonlinear damper, the hysteretic friction tuned inerter damper (HFTID), for seismic control of engineering structures. The HFTID consists of a tuned inerter damper (TID) in parallel with a hysteresis spring friction element. The analytical expressions for the force and displacement transmissibility of a SDOF system with HFTID are derived via the method of harmonic balance, and the solutions are validated through numerical simulation based on the OpenSees. Parametric study is carried out to investigate the influence of the hysteresis spring friction element on the transmissibility. The optimal tuning parameters of HFTID are obtained and optimality corresponds to minimization of the maximum force or displacement transmissibility. By applying a sequence of curve fitting scheme, design formula for optimal tuning parameters of the HFTID are formulated, which are expressed as functions of structural damping ratio, inertance–mass ratio, and stiffness ratio. The vibration reduction effect and robustness of vibration control systems with HFTID and TID are compared. The results demonstrate that the HFTID can further reduce the transmissibility of vibration control system compared with TID. Moreover, HFTID is more robust than the TID in the case of detuning. Finally, a multi‐story building is taken as an example to demonstrate the effectiveness and practicability of the HFTID in reducing seismic response.