  The spin-lattice relaxation rate $T_1^{-1}$ of $^1$H-NMR has been measured in
(CH$_3$)$_2$CHNH$_3$Cu(Cl$_x$Br$_{1-x}$)$_3$ with $x=0$ and 0.35, in order to
investigate the microscopic magnetism of systems. Previous macroscopic
magnetization and specific heat measurements suggested that these two exist in
a singlet-dimer phase. The temperature dependence of $T_1^{-1}$ in an $x=0$
system decreased exponentially toward zero, indicating microscopic evidence of
the gapped singlet ground state, which is consistent with the macroscopic
experiments. At the same time, in the $x=0.35$ system, $T_1^{-1}$ showed a
sharp peak structure at around 7.5 K though no splitting of $^1$H-NMR spectra
indicative of the magnetic ordering was observed. We discuss the observed sharp
peak structure in the $x=0.35$ system with the soft mode toward the exotic
magnetic ground state suggested by the recent $\mu$SR experiments.
