We demonstrate that the plasmon in one-dimensional Coulomb interacting electron fluids can develop a finite-momentum maxon-roton-like nonmonotonic energy-momentum dispersion. Such an unusual nonmonotonicity arises from the strongly interacting 1/r Coulomb potential going beyond the conventional band linearization approximation used in the standard bosonization theories of Luttinger liquids. We provide details for the nonmonotonic plasmon dispersion using both bosonization and random-phase approximation theories. We also calculate the specific heat including the nonmonotonicity and discuss possibilities for observing the nonmonotonic plasmon dispersion in various physical systems, including semiconductor quantum wires, carbon nanotubes, and the twisted bilayer graphene at subdegree twist angles, which naturally realize one-dimensional domain-wall states. We provide results for several different models of long-range interaction showing that the nonomonotonic charge collective mode dispersion is a generic phenomenon in one-dimensional strongly interacting electron systems.