For positive integers $r > \ell \geq 1$, an $\ell$-cycle in an $r$-uniform hypergraph is a cycle where each edge consists of $r$ vertices and each pair of consecutive edges intersect in $\ell$ vertices. For $\ell \geq 2$, we determine the limiting distribution of the number of Hamilton $\ell$-cycles in an Erdős--Rényi random hypergraph. The behavior is distinguished in two cases: -When $\ell \geq 3$, the number of cycles concentrates when the expectation diverges and converges to a Poisson distribution when the expectation is constant. -When $\ell = 2$, the normalized number of cycles converges to a lognormal distribution when the expectation diverges and converges to a lognormal mixture of Poisson distributions when the expectation is constant. As a result we pin down the exact threshold for the appearance of non-linear Hamilton cycles in random hypergraphs, confirming a conjecture of Narayanan and Schacht.