  We report the results of magnetic susceptibility $\chi$ and $^1$H nuclear
magnetic resonance (NMR) measurements on a three-dimensional hyper-honeycomb
lattice compound [(C$_2$H$_5$)$_3$NH]$_2$Cu$_2$(C$_2$O$_4$)$_3$ (CCCO). The
average value of the antiferromagnetic (AFM) exchange coupling between the
Cu$^{2+}$ ($S$ = 1/2) spins was determined to be $J$~$\sim$~50 K from the
$\chi$ measurements. No long-range magnetic ordering has been observed down to
$T$ = 50 mK, although NMR lines become slightly broader at low temperatures
below 1 K. The broadening of the NMR spectrum observed below 1 K reveals that
the Cu spin moments remain at this temperature, suggesting a non-spin-singlet
ground state. The temperature and magnetic field dependence of 1/$T_1$ at
temperatures above 20 K is well explained by paramagnetic thermal spin
fluctuations where the fluctuation frequency of Cu$^{2+}$ spins is higher than
the NMR frequency of the order of MHz. However, a clear signature of the
slowing down of the Cu$^{2+}$ spin fluctuations was observed at low
temperatures where 1/$T_1$ shows a thermally-activated behavior. The magnetic
field dependence of the magnitude of the spin excitation gap suggests that the
magnetic behaviors of CCCO are characterized as an AFM chain at low
temperatures.
