Context. Irregular dwarf galaxies of the Local Group have very varied properties and star formation histories. Some of them formed the majority of their stars very late compared to others. Extreme examples of this are Leo A and Aquarius, which reached the peak of star formation at z < 1 (more than 6 Gyr after the Big Bang). This fact seemingly challenges the ΛCDM cosmological framework because the dark matter halos of these galaxies on average should assemble the majority of their masses before z ~ 2 (<3 Gyr after the Big Bang). Aims. We investigate whether the delayed star formation histories of some irregular dwarf galaxies might be explained purely by the stochasticity of their mass assembly histories coupled with the effect of cosmic reionization. Methods. We developed a semi-analytic model to follow the accretion of baryonic matter, star formation, and stellar feedback in dark matter halos with present-day virial masses 109 M⊙ < Mdm,0 < 1011 M⊙ and with different stochastic growth histories obtained using the PINOCCHIO code based on Lagrangian perturbation theory. Results. We obtain the distributions of observable parameters and the evolution histories for these galaxies. Accretion of baryonic matter is strongly suppressed after the epoch of reionization in some models, but the galaxies continue to accrete dark matter and eventually reach enough mass for accretion of baryonic matter to begin again. These “reborn” model galaxies show delayed star formation histories that are very similar to those of Leo A and Aquarius. Conclusions. We find that the stochasticity caused by mass assembly histories is enhanced in systems with virial masses ~1010 M⊙ because of their sensitivity to the photoionizing intergalactic radiation field after the epoch of reionization. This results in qualitatively different star formation histories in late- and early-forming galaxies, and it might explain the peculiar star formation histories of irregular dwarf galaxies such as Leo A and Aquarius.