  Strong lensed quasi-stellar objects (QSOs) are valuable probes of the
universe in numerous aspects. Two of these applications, reverberation mapping
and measuring time delays for determining cosmological parameters, require the
source QSOs to be variable with sufficient amplitude. In this paper, we
forecast the number of strong lensed QSOs with sufficient variability to be
detected by the Vera C. Rubin Telescope Legacy Survey of Space and Time (LSST).
The damped random walk model is employed to model the variability amplitude of
lensed QSOs taken from a mock catalog by Oguri & Marshall (2010). We expect
30--40% of the mock lensed QSO sample, which corresponds to $\sim$1000, to
exhibit variability detectable with LSST. A smaller subsample of 250 lensed
QSOs will show larger variability of $>0.15$~mag for bright lensed images with
$i<21$ mag, allowing for monitoring with smaller telescopes. We discuss
systematic uncertainties in the prediction by considering alternative
prescriptions for variability and mock lens catalog with respect to our
fiducial model. Our study shows that a large-scale survey of lensed QSOs can be
conducted for reverberation mapping and time delay measurements following up on
LSST.
