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.
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