We report the spectroscopic confirmation and fundamental properties of TOI-757 b, a mini-Neptune on a 17.5-d orbit transiting a bright star (V = 9.7 mag) discovered by the TESS mission. We acquired high-precision radial velocity measurements with the HARPS, ESPRESSO, and PFS spectrographs to confirm the planet detection and determine its mass. We also acquired space-borne transit photometry with the CHEOPS space telescope to place stronger constraints on the planet radius, supported with ground-based LCOGT photometry. WASP and KELT photometry were used to help constrain the stellar rotation period. We also determined the fundamental parameters of the host star. We find that TOI-757 b has a radius of R-p = 2.5 +/- 0.1 R-circle plus and a mass of M-p = 10.5(-2.1)(+2.2) M-circle plus, implying a bulk density of rho(p) = 3.6 +/- 0.8 g cm(-3). Our internal composition modelling was unable to constrain the composition of TOI-757 b, highlighting the importance of atmospheric observations for the system. We also find the planet to be highly eccentric with e = 0.39(-0.07)(+0.08), making it one of the very few highly eccentric planets among precisely characterized mini-Neptunes. Based on comparisons to other similar eccentric systems, we find a likely scenario for TOI-757 b's formation to be high eccentricity migration due to a distant outer companion. We additionally propose the possibility of a more intrinsic explanation for the high eccentricity due to star-star interactions during the earlier epoch of the Galactic disc formation, given the low metallicity and older age of TOI-757.
