NMR measurements of the Si-29 spin-lattice relaxation time T-1 were used to probe the spin-1/2 random Heisenberg chain compound BaCu2(Si1-xGex)(2)O-7. Remarkable differences between the pure (x = 0) and the fully random (x = 0.5) cases are observed, indicating that randomness generates a distribution of local magnetic relaxations. This distribution, which is reflected in a stretched exponential NMR relaxation, exhibits a progressive broadening with decreasing temperature, caused by a growing inequivalence of magnetic sites. Compelling independent evidence for the influence of randomness is also obtained from magnetization data and Monte Carlo calculations. These results suggest the formation of random-singlet states in this class of materials, as previously predicted by theory.