Context. Although many studies have been performed so far, there are still dozens of low-mass stars and brown dwarfs in the young σ Orionis open cluster without detailed spectroscopic characterisation. Aims: We look for unknown strong accretors and disc hosts that were undetected in previous surveys. Methods: We collected low-resolution spectroscopy (R ~ 700) of ten low-mass stars and brown dwarfs in σ Orionis with OSIRIS at the Gran Telescopio Canarias under very poor weather conditions. These objects display variability in the optical, infrared, Hα, and/or X-rays on time scales of hours to years. We complemented our spectra with optical and near-/mid-infrared photometry. Results: For seven targets, we detected lithium in absorption, identified Hα, the calcium doublet, and forbidden lines in emission, and/or determined spectral types for the first time. We characterise in detail a faint, T Tauri-like brown dwarf with an 18 h-period variability in the optical and a large Hα equivalent width of -125 ± 15 Å, as well as two M1-type, X-ray-flaring, low-mass stars, one with a warm disc and forbidden emission lines, the other with a previously unknown cold disc with a large inner hole. Conclusions: New unrevealed strong accretors and disc hosts, even below the substellar limit, await discovery among the list of known σ Orionis stars and brown dwarfs that are variable in the optical and have no detailed spectroscopic characterisation yet.