In fluid film lubrication investigations, the homogenized Reynolds equation is used as an averaging model to deal with microstructures induced by rough or textured surfaces. The objective is a reduction of computation time compared to directly solving the original Reynolds equation which would require very fine computational grids. By solving cell problems on the microscale, homogenized coefficients are computed to set up a homogenized problem on the macroscale. For the latter, the discretization can be chosen much coarser than for the original Reynolds equation. However, the microscale cell problems depend on the macroscale film thickness and thus become parameter-dependent. This requires a large number of cell problems to be solved, contradicting the objective of accelerating simulations. A reduced basis method is proposed which significantly speeds up the solution of the cell problems and the computation of the homogenized coefficients without loss of accuracy. The suitability of both the homogenization technique and the combined homogenization/reduced basis method is documented for the application to textured journal bearings. For this purpose, numerical results are presented where deviations from direct solutions of the original Reynolds equation are investigated and the reduction of computational cost is measured.