  We report the spectroscopic confirmation and fundamental properties of
TOI-757 b, a mini-Neptune on a 17.5-day 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 $R_{\mathrm{p}} = 2.5 \pm 0.1
R_{\oplus}$ and a mass of $M_{\mathrm{p}} = 10.5^{+2.2}_{-2.1} M_{\oplus}$,
implying a bulk density of $\rho_{\text{p}} = 3.6 \pm 0.8$ g cm$^{-3}$. Our
internal composition modeling 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$^{+0.08}_{-0.07}$, 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 disk formation, given the low metallicity and older age of TOI-757.
