Abstract This is the first paper in a series aimed at modeling the black hole (BH) mass function, from the stellar to the intermediate to the (super)massive regime. In the present work, we focus on stellar BHs and provide an ab initio computation of their mass function across cosmic times; we mainly consider the standard, and likely dominant, production channel of stellar-mass BHs constituted by isolated single/binary star evolution. Specifically, we exploit the state-of-the-art stellar and binary evolutionary code SEVN, and couple its outputs with redshift-dependent galaxy statistics and empirical scaling relations involving galaxy metallicity, star formation rate and stellar mass. The resulting relic mass function dN / dVd log m • as a function of the BH mass m _• features a rather flat shape up to m _• ≈ 50 M _⊙ and then a log-normal decline for larger masses, while its overall normalization at a given mass increases with decreasing redshift. We highlight the contribution to the local mass function from isolated stars evolving into BHs and from binary stellar systems ending up in single or binary BHs. We also include the distortion on the mass function induced by binary BH mergers, finding that it has a minor effect at the high-mass end. We estimate a local stellar BH relic mass density of ρ _• ≈ 5 × 10⁷ M _⊙ Mpc⁻³, which exceeds by more than two orders of magnitude that in supermassive BHs; this translates into an energy density parameter Ω_• ≈ 4 × 10⁻⁴, implying that the total mass in stellar BHs amounts to ≲1% of the local baryonic matter. We show how our mass function for merging BH binaries compares with the recent estimates from gravitational wave observations by LIGO/Virgo, and discuss the possible implications for dynamical formation of BH binaries in dense environments like star clusters. We address the impact of adopting different binary stellar evolution codes (SEVN and COSMIC) on the mass function, and find the main differences to occur at the high-mass end, in connection with the numerical treatment of stellar binary evolution effects. We highlight that our results can provide a firm theoretical basis for a physically motivated light seed distribution at high redshift, to be implemented in semi-analytic and numerical models of BH formation and evolution. Finally, we stress that the present work can constitute a starting point to investigate the origin of heavy seeds and the growth of (super)massive BHs in high-redshift star-forming galaxies, that we will pursue in forthcoming papers.