We analyze the scaling properties of the diurnal variation of galactic cosmic rays (GCRs) in Solar Cycle 24 and the solar minima between Solar Cycles 23/24 and 24/25 for 2007 – 2019 based on the count rates of the Oulu, Newark, Hermanus, and Potchefstroom neutron monitors. The scaling features of the GCR diurnal variation are studied by evaluating the Hurst exponent, a quantitative parameter used as an indicator of the state of the randomness of a time series. We estimate the Hurst exponents for GCR diurnal-variation parameters amplitude and phase using structure-function and detrended-fluctuation-analysis methods. Results show that the Hurst exponents for the GCR diurnal variation vary in the range from $\approx0.3$
≈
0.3
to $\approx0.9$
≈
0.9
, with a general tendency of being systematically above 0.5. It suggests that the GCR diurnal variation reveals a more persistent structure than Brownian motion. However, the time series of GCR diurnal-variation amplitude and phase evolve from a more persistent structure in the solar minimum between Solar Cycles 23/24 in 2007 – 2009 to a more random character in and near the solar maximum 2012 – 2014. This observation seems to be in agreement with the general configuration of the heliosphere through the 11-year solar-activity cycle. Moreover, the temporal profile of the Hurst exponent for GCR diurnal amplitude and phase around the beginning of the solar minimum between Solar Cycles 24/25 (2018 – 2019) differs from the solar minimum between Solar Cycles 23/24 in 2007 – 2009, suggesting a dependence on solar-magnetic polarity. These findings could shed more light on GCR particle transport in the turbulent heliosphere over the solar cycle.