The HeH+ molecule is the first to be formed in the Universe. Its recent detection, in the interstellar medium, has increased the interest in the study of the physical and chemical properties of this ion. Here, we report exact quantum time-independent calculations of the collisional cross sections and rate coefficients for the rotational excitation of HeH+ by H. Reactive and exchange channels are taken into account in the scattering calculations. Cross sections are computed for energies of up to 10 000 cm-1, enabling the computation of rate coefficients for temperatures of up to 500 K. The strongest collision-induced rotational HeH+ transitions are those with Δj = 1. Previous results obtained using approximate treatment are compared to the new ones, and significant differences are found. The new rate coefficients are also compared to those for electron-impact rotational excitation, and we found that collisions with H dominate the excitation of HeH+ in media where the electron fraction is less than 10-4. In the light of those results, we recommend the use of the new HeH+-H collisional data in order to accurately model HeH+ excitation in both the interstellar media and early Universe.