  We determine the quantum Cram\'er-Rao bound for the precision with which the
oscillator frequency and damping constant of a damped quantum harmonic
oscillator in an arbitrary Gaussian state can be estimated. This goes beyond
standard quantum parameter estimation of a single mode Gaussian state for which
typically a mode of fixed frequency is assumed. We present a scheme through
which the frequency estimation can nevertheless be based on the known results
for single-mode quantum parameter estimation with Gaussian states. Based on
these results, we investigate the optimal measurement time. For measuring the
oscillator frequency, our results unify previously known partial results and
constitute an explicit solution for a general single-mode Gaussian state.
Furthermore, we show that with existing carbon nanotube resonators (see J.
Chaste et al.~Nature Nanotechnology 7, 301 (2012)) it should be possible to
achieve a mass sensitivity of the order of an electron mass $\text{Hz}^{-1/2}$.
