The effects of a Wigner molecule on the current noise and conductance of a one-dimensional quantum dot with two electrons are investigated. Focusing on a lateral transport setup, the sequential regime is considered. Tunnelling rates through the dot are evaluated within an exact diagonalisation scheme. They strongly depend on electron interactions, showing a markedly different behaviour in the presence of a Wigner molecule with respect to the weak interactions case, and thus modify the transport and current noise and the dot. For weak interactions negative differential conductance and super-Poissonian noise are found. As interactions increase, a Wigner molecule develops: it suppresses the negative differential conductance and turns the shot noise to sub-Poissonian values. In particular, the noise is found to be a sensitive probe of the Wigner molecule.