We construct a holographic model for dark energy in the Brans–Dicke cosmology by using the holographic principle considering the Barrow entropy instead of the standard Bekenstein–Hawking one. The former arises from the effort to account for quantum gravitational effects in black hole physics and, according to the gravity–thermodynamic conjecture, in the cosmological framework. In order to explore the cosmological consequences of our model, we consider the Hubble horizon as the IR cutoff. We investigate both the non-interacting and interacting cases with the sign-changeable and linear interactions, showing that they can explain the present accelerated phase of the Universe expansion, in contrast to the standard holographic dark energy model. We then perform the stability analysis according to the squared sound speed. We find that, while the non-interacting model is unstable against small perturbations, the sign-changeable interacting one can be stable only for suitable values of the model parameters. On the other hand, the linear interacting model always predicts a stable Universe. The consistency of the model with respect to cosmological observations is discussed.