The celebrated canonical Ramsey theorem of Erdős and Rado implies that for a given $k$-uniform hypergraph (or $k$-graph) $H$, if $n$ is sufficiently large then any colouring of the edges of the complete $k$-graph $K^{(k)}_n$ gives rise to copies of $H$ that exhibit certain colour patterns. We are interested in sparse random versions of this result and the threshold at which the random $k$-graph ${\mathbf{G}}^{(k)}(n,p)$ inherits the canonical Ramsey properties of $K^{(k)}_n$. Our main result here pins down this threshold when we focus on colourings that are constrained by some prefixed lists. This result is applied in an accompanying work of the authors on the threshold for the canonical Ramsey property (with no list constraints) in the case that $H$ is a (2-uniform) even cycle.