The structure, lattice dynamics and mechanical properties of magnesium hydroxide have been investigated by static density functional theory calculations as well as ab initio molecular dynamics. The hypothesis of a superstructure existing in the lattice formed by the hydrogen atoms has been tested. The elastic constants of the material have been calculated with a static deformations approach and are in fair agreement with the experimental data. The hydrogen subsystem structure exhibits signs of disordered behaviour while maintaining correlations between the angular positions of neighbouring atoms. We establish that the essential angular correlations between hydrogen positions are maintained to a temperature of at least 150 K and that they are well described by a physically motivated probabilistic model. The rotational degree of freedom appears to be decoupled from the lattice directions above 30 K.