Cylindrical contact homology is a comparatively simple incarnation of symplectic field theory whose existence and invariance under suitable hypotheses was recently established by Hutchings and Nelson. We study this invariant for a general Brieskorn 3-manifold \(\Sigma(a_1,\ldots, a_n)\), and give a complete description of the cylindrical contact homology for this 3-manifold equipped with its natural contact structure, for any \(a_j\) satisfying \(\frac{1}{a_1} + \cdots + \frac{1}{a_n} < n-2\).