  The discovery of the electromagnetic counterpart to the binary neutron star
merger GW170817 has opened the era of gravitational-wave multi-messenger
astronomy. Rapid identification of the optical/infrared kilonova enabled a
precise localization of the source, which paved the way to deep
multi-wavelength follow-up and its myriad of related science results. Fully
exploiting this new territory of exploration requires the acquisition of
electromagnetic data from samples of neutron star mergers and other
gravitational wave sources. After GW170817, the frontier is now to map the
diversity of kilonova properties and provide more stringent constraints on the
Hubble constant, and enable new tests of fundamental physics. The Vera C. Rubin
Observatory's Legacy Survey of Space and Time (LSST) can play a key role in
this field in the 2020s, when an improved network of gravitational-wave
detectors is expected to reach a sensitivity that will enable the discovery of
a high rate of merger events involving neutron stars (about tens per year) out
to distances of several hundred Mpc. We design comprehensive
target-of-opportunity observing strategies for follow-up of gravitational-wave
triggers that will make the Rubin Observatory the premier instrument for
discovery and early characterization of neutron star and other compact object
mergers, and yet unknown classes of gravitational wave events.
