Aims. We investigated the X-ray emission from young stars and brown dwarfs in the σ Orionis cluster (τ ∼ 3 Ma, d ∼ 385 pc) and its relation to mass, the presence of circumstellar discs, and separation to the cluster centre by taking advantage of the superb spatial resolution of the Chandra X-ray Observatory. Methods. We used public HRC-I/Chandra data from a 97.6 ks pointing towards the cluster centre and complemented them with X-ray data from IPC/Einstein, HRI/ROSAT, EPIC/XMM-Newton, and ACIS-S/Chandra together with optical and infrared photometry and spectroscopy from the literature and public catalogues. On our HRC-I/Chandra data, we measured count rates, estimated X-ray fluxes, and searched for short-term variability. We also looked for long-term variability by comparing with previous X-ray observations. Results. Among the 107 detected X-ray sources, there were 70 cluster stars with known signposts of youth, two young brown dwarfs, 12 cluster member candidates, four field dwarfs, and two galaxies with optical-infrared counterpart. The remaining sources were extragalactic. Based on a robust Poisson-χ 2 analysis, nine cluster stars displayed flares or rotational modulation during the HRC-I observations, while eight other stars and one brown dwarf showed X-ray flux variations between the HRC-I and IPC, HRI, and EPIC epochs. We constructed a cluster X-ray luminosity function from O9.5 (about 18 M ) to M6.5 (about 0.06 M ). We found: (i) that early-type stars in multiple systems or with spectroscopic peculiarities tend to display X-ray emission; (ii) that the two detected brown dwarfs and the least-massive star are among the σ Orionis objects with the highest L X /L J ratios; and (iii) that a large fraction of known classical T Tauri stars in the cluster are absent in this and other X-ray surveys. Finally, from a spatial distribution analysis, we quantified the impact of sensitivity degradation towards the HRC-I borders on the detection of faint X-ray sources and concluded that dozens X-ray σ Orionis stars and brown dwarfs still need to be detected.