Context. Large sub-Neptunes are uncommon around the coolest stars in the Galaxy and are rarer still around those that are metal-poor. However, owing to the large planet-to-star radius ratio, these planets are highly suitable for atmospheric study via transmission spectroscopy in the infrared, such as with JWST. Aims. Here we report the discovery and validation of a sub-Neptune orbiting the thick-disk, mid-M dwarf star TOI-2406. The star's low metallicity and the relatively large size and short period of the planet make TOI-2406 b an unusual outcome of planet formation, and its characterisation provides an important observational constraint for formation models. Methods. We first infer properties of the host star by analysing the star's near-infrared spectrum, spectral energy distribution, and Gaia parallax. We use multi-band photometry to confirm that the transit event is on-target and achromatic, and we statistically validate the TESS signal as a transiting exoplanet. We then determine physical properties of the planet through global transit modelling of the TESS and ground-based time-series data. Results. We determine the host to be a metal-poor M4 V star, located at a distance of 56 pc, with properties Teff = 3100 ± 75 K, M∗ = 0.162 ± 0.008Mo˙, R∗ = 0.202 ± 0.011Ro˙, and [Fe∕ H] = -0.38 ± 0.07, and a member of the thick disk. The planet is a relatively large sub-Neptune for the M-dwarf planet population, with Rp = 2.94 ± 0.17R⊕ and P= 3.077 d, producing transits of 2% depth. We note the orbit has a non-zero eccentricity to 3σ, prompting questions about the dynamical history of the system. Conclusions. This system is an interesting outcome of planet formation and presents a benchmark for large-planet formation around metal-poor, low-mass stars. The system warrants further study, in particular radial velocity follow-up to determine the planet mass and constrain possible bound companions. Furthermore, TOI-2406 b is a good target for future atmospheric study through transmission spectroscopy. Although the planet's mass remains to be constrained, we estimate the S/N using amass-radius relationship, ranking the system fifth in the population of large sub-Neptunes, with TOI-2406 b having a much lower equilibrium temperature than other spectroscopically accessible members of this population.