Due to the variability of raw materials and the fluctuation in the manufacturing process, degradation of products may exhibit unit-to-unit variability in a population. The heterogeneous degradation rates can be viewed as random effects, which are often modeled by a normal distribution. Despite of its mathematical convenience, the normal distribution has certain limitations in modeling the random effects. In this study, we propose a novel random-effects Wiener process model based on ideas from accelerated failure time principle. An inverse Gaussian (IG) distribution can be used to characterize the unit-specific heterogeneity in degradation paths, which overcomes the disadvantages of the traditional models and provides more flexibility in the degradation modeling using Wiener processes. Properties of the model are investigated, and statistical inference based on the maximum likelihood estimation and the EM algorithm is established. An extension of the model to the constant-stress accelerated degradation test (ADT) is developed. The effectiveness and applicability of the proposed model are validated using a laser degradation dataset and an LED ADT dataset.