A stress equilibration procedure for hyperelastic material models is proposed and analyzed in this paper. Based on the displacement‐pressure approximation computed with a stable finite element pair, it constructs, in a vertex‐patch‐wise manner, an H(div)‐conforming approximation to the first Piola‐Kirchhoff stress. This is done in such a way that its associated Cauchy stress is weakly symmetric in the sense that its antisymmetric part is zero tested against continuous piecewise linear functions. Our main result is the identification of the subspace of test functions perpendicular to the range of the local equilibration system on each patch which turn out to be rigid body modes associated with the current configuration. Momentum balance properties are investigated analytically and numerically and the resulting stress reconstruction is shown to provide improved results for surface traction forces by computational experiments.