  Tension between cosmic microwave background-based and distance ladder-based
determinations of the Hubble constant ${\rm H}_{\rm 0}$ motivates pursuit of
independent methods that are not subject to the same systematic effects. A
promising alternative, proposed by Refsdal in 1964, relies on the inverse
scaling of ${\rm H}_{\rm 0}$ with the delay between the arrival times of at
least two images of a strongly-lensed variable source such as a quasar. To
date, Refsdal's method has mostly been applied to quasars lensed by individual
galaxies rather than by galaxy clusters. Using the three quasars strongly
lensed by galaxy clusters (SDSS J1004+4112, SDSS J1029+2623, and SDSS
J2222+2745) that have both multiband Hubble Space Telescope data and published
time delay measurements, we derive ${\rm H}_{\rm 0}$, accounting for the
systematic and statistical sources of uncertainty. While a single time delay
measurement does not yield a well-constrained ${\rm H}_{\rm 0}$ value,
analyzing the systems together tightens the constraint. Combining the six time
delays measured in the three cluster-lensed quasars gives ${\rm H}_{\rm 0}$ =
74.1 $\pm$ 8.0 km s$^{-1}$ Mpc$^{-1}$. To reach 1$\%$ uncertainty in ${\rm
H}_{\rm 0}$, we estimate that a sample size of order of 620 time delay
measurements of similar quality as those from SDSS J1004+4112, SDSS J1029+2623,
and SDSS J2222+2745 would be needed. Improving the lens modeling uncertainties
by a factor of two and a half may reduce the needed sample size to 100 time
delays, potentially reachable in the next decade.
