A light vector boson, Z(d), associated with a "dark sector" U(1)(d) gauge group has been introduced to explain certain astrophysical observations as well as low energy laboratory anomalies. In such models, the Higgs boson may decay into X + Z(d), where X = Z, Z(d) or gamma. Here, we provide estimates of those decay rates as functions of the Z(d) coupling through either mass mixing (e.g., via an enlarged Higgs mechanism) or through heavy new fermion loops and examine the implied LHC phenomenology. Our studies focus on the higher m(Zd) case, greater than or similar to several GeV, where the rates are potentially measurable at the LHC, for interesting regions of parameter spaces, at a level complementary to low energy experimental searches for the Z(d). We also show how measurement of the Z(d) polarization (longitudinal vs transverse) can be used to distinguish the physics underlying these rare decays.