Theory and stochastic lattice-gas modeling is developed for the formation of intercalated metal islands in the gallery between the top layer and the underlying layer at the surface of layered materials. Our model for this process involves deposition of atoms, some fraction of which then enter the gallery through well-separated point-like defects in the top layer. Subsequently, these atoms diffuse within the subsurface gallery leading to nucleation and growth of intercalated islands nearby the defect point source. For the case of a single point defect, continuum diffusion equation analysis provides insight into the nucleation kinetics. However, complementary tailored lattice-gas modeling produces a more comprehensive and quantitative characterization. We analyze the large spread in nucleation times and positions relative to the defect for the first nucleated island. We also consider the formation of subsequent islands and the evolution of island growth shapes. The shapes reflect in part our natural adoption of a hexagonal close-packed island structure. Motivation and support for the model is provided by scanning tunneling microscopy observations of the formation of intercalated metal islands in highly-ordered pyrolytic graphite at higher temperatures.