  We propose a long-term strategy for detecting thermal neutrinos from the
remnant of binary-neutron-star mergers with a future M-ton water-Cherenkov
detector such as Hyper-Kamiokande. Monitoring >~2500 mergers within <~200 Mpc,
we may be able to detect a single neutrino with a human-time-scale operation of
~80 Mt years for the merger rate of 1 Mpc^{-3} Myr^{-1}, which is slightly
lower than the median value derived by the LIGO-Virgo collaboration with
GW170817. Although the number of neutrino events is minimal, contamination from
other sources of neutrinos can be reduced efficiently to ~0.03 by analyzing
only ~1 s after each merger identified with gravitational-wave detectors if
Gadolinium is dissolved in the water. The contamination may be reduced further
to ~0.01 if we allow the increase of waiting time by a factor of ~1.7. The
detection of even a single neutrino can pin down the energy scale of thermal
neutrino emission from binary-neutron-star mergers and could strongly support
or disfavor formation of remnant massive neutron stars. Because the dispersion
relation of gravitational waves are now securely constrained to that of
massless particles with a corresponding limit on the graviton mass of
<~10^{-22} eV/c^2 by binary-black-hole mergers, the time delay of a neutrino
from gravitational waves can be used to put an upper limit of <~O(10) meV/c^2
on the absolute neutrino mass in the lightest eigenstate. Large neutrino
detectors will enhance the detectability, and in particular, 5 Mt Deep-TITAND
and 10 Mt MICA planned in the future will allow us to detect thermal neutrinos
every ~16 and 8 years, respectively, increasing the significance.
