In this note we study the emergence of Hamiltonian Berge cycles in random r-uniform hypergraphs. For we prove an optimal stopping time result that if edges are sequentially added to an initially empty r-graph, then as soon as the minimum degree is at least 2, the hypergraph with high probability has such a cycle. In particular, this determines the threshold probability for Berge Hamiltonicity of the Erdős–Rényi random r-graph, and we also show that the 2-out random r-graph with high probability has such a cycle. We obtain similar results for weak Berge cycles as well, thus resolving a conjecture of Poole.