  The discovery of a binary neutron star merger (NSM) through both its
gravitational wave and electromagnetic emission has revealed these events to be
key sites of r-process nucleosynthesis. Here, we evaluate the prospects of
finding the remnants of Galactic NSMs by detecting the gamma-ray decay lines
from their radioactive r-process ejecta. We find that $^{126}$Sn, which has
several lines in the energy range 415-695 keV and resides close to the second
r-process peak, is the most promising isotope, because of its half-life
$t_{1/2}=2.30(14)\times 10^{5}$ yr being comparable to the ages of recent NSMs.
Using a Monte Carlo procedure, we predict that multiple remnants are detectable
as individual sources by next-generation gamma-ray telescopes which achieve
sub-MeV line sensitivities of $\sim 10^{-8}$-$10^{-6}$ $\gamma$ cm$^{-2}$
s$^{-1}$. However, given the unknown locations of the remnants, the most
promising search strategy is a systematic survey of the Galactic plane and
bulge extending to high Galactic latitudes. Individual known supernova remnants
which may be mis-classified NSM remnants could also be targeted, especially
those located outside the Galactic plane. Detection of a moderate sample of
Galactic NSM remnants would provide important clues to unresolved issues such
as the production of actinides in NSMs, properties of merging NS binaries, and
even help distinguish them from rare supernovae as current Galactic r-process
sources. We also investigate the diffuse flux from longer-lived nuclei (e.g.
$^{182}$Hf) that could in principle trace the Galactic spatial distribution of
NSMs over longer timescales, but find that the detection of the diffuse flux
appears challenging even with next-generation telescopes.
