The spin-lattice relaxation of protons in NH4ClO4 at low temperatures has been studied theoretically and experimentally. The NH4 librational tunneling determines the spin-librational wavefunctions, which are derived first. The dominant transition rates related to the magnetic dipolar interaction between the NH4 protons are then obtained. They reproduce well the angular dependence of the proton relaxation time. When all the transition rates are taken into account, both the temperature and frequency dependence of the relaxation rate agree with our experimental data for a powder sample. Reasons for the non-exponential relaxation are discussed. Apart from the angular dependence of the relaxation rate, they include the tunnel frequency distribution, the coupling to the tunnel reservoir and the effect of deuterons in their natural concentration.