The effect of intensity fluctuations on laser-induced coupling of electronic and vibrational states of a trapped ion is studied analytically. Different types of noise relevant to the experiments are considered. The resulting decoherence phenomenology is shown to present nontrivial characteristics. Noise color leads to nonexponential decay of the coherences: after the initial decay, determined by the probability distribution, a transient period specific to the noise spectrum is apparent; at large times, exponential decay sets in for widely different noise properties. The detection of these features in the evolution of an initial coherent state of the vibrational mode is discussed.