The concept of spaser as the coherent near-field generator and nanolaser based on nanoscale plasmonic resonators has been successfully demonstrated in number of experiments. Here we have developed the theoretical framework for the basic linewidth description of these active plasmonic structures and, in particular, linewidth enhancement - additional line broadening due to the resonator noise. In order to achieve this, we have introduced explicitly the time dependence in the quasistatic description of localized surface plasmon resonances via inclusion of the dispersion of a spectral parameter defining the resonant frequency. Linewidth enhancement factor was estimated for semiconductor gain medium and was found to be of order of 3 to 6, strongly depending on carrier density in the active layer, and resulting in more than order of magnitude broader linewidth compared to that, predicted by the Schawlow-Townes theory.