We present new developments for an ab-initio model of the neutron relative biological effectiveness (RBE) in inducing specific classes of DNA damage. RBE is evaluated as a function of the incident neutron energy and of the depth inside a human-sized phantom. The adopted mechanistic approach traces neutron RBE back to its origin, i.e. neutron physical interactions with biological tissues. To this aim, we combined the simulation of radiation transport through biological matter, performed with the Monte Carlo code PHITS, and the prediction of DNA damage using analytical formulas, which ground on a large database of biophysical radiation track structure simulations performed with the code PARTRAC. In particular, two classes of DNA damage were considered: sites and clusters of double strand breaks (DSBs), which are known to be correlated with cell fate following radiation exposure. Within a coherent modelling framework, this approach tackles the variation of neutron RBE in a wide energy range, from thermal neutrons to neutrons of hundreds of GeV, and reproduce effects related to depth in a human-size receptor, as well as to the receptor size. RBE predictions were successfully compared to the currently adopted radiation protection standards for neutron weighting factors. Our results also suggest that great care is needed when applying weighting factors as a function of incident neutron energy, not explicitly considering RBE variation in the target. Look-up RBE tables and an analytical representation of the maximal RBE vs. neutron energy are finally proposed, to facilitate the use of our results in radiation biology studies and radiation protection applications with neutron exposures.