  Optically luminous quasars at $z > 5$ are important probes of super-massive
black hole (SMBH) formation. With new and future radio facilities, the
discovery of the brightest low-frequency radio sources in this epoch would be
an important new probe of cosmic reionization through 21-cm absorption
experiments. In this work, we systematically study the low-frequency radio
properties of a sample of 115 known spectroscopically confirmed $z>5$ quasars
using the second data release of the Low Frequency Array (LOFAR) Two Metre Sky
survey (LoTSS-DR2), reaching noise levels of $\sim$80 $\mu$Jy beam$^{-1}$ (at
144 MHz) over an area of $\sim5720$ deg$^2$. We find that 41 sources (36%) are
detected in LoTSS-DR2 at $>2 \sigma$ significance and we explore the evolution
of their radio properties (power, spectral index, and radio loudness) as a
function of redshift and rest-frame ultra-violet properties. We obtain a median
spectral index of $-0.29^{+0.10}_{-0.09}$ by stacking 93 quasars using
LoTSS-DR2 and Faint Images of the Radio Sky at Twenty Centimetres (FIRST) data
at 1.4 GHz, in line with observations of quasars at $z<3$. We compare the radio
loudness of the high-$z$ quasar sample to a lower-$z$ quasar sample at $z\sim2$
and find that the two radio loudness distributions are consistent with no
evolution, although the low number of high-z quasars means that we cannot rule
out weak evolution. Furthermore, we make a first order empirical estimate of
the $z=6$ quasar radio luminosity function, which is used to derive the
expected number of high-$z$ sources that will be detected in the completed
LoTSS survey. This work highlights the fact that new deep radio observations
can be a valuable tool in selecting high-$z$ quasar candidates for follow-up
spectroscopic observations by decreasing contamination of stellar dwarfs and
reducing possible selection biases introduced by strict colour cuts.
