The question of the basis property of a system of eigenfunctions of one spectral problem for a discontinuous second-order differential operator with a spectral parameter under discontinuity conditions is considered in the weighted grand-Lebesgue spaces L p),ρ (0, 1), 1 < p < +∞, with a general weight ρ(·). These spaces are non-separable and therefore it is necessary to define its subspace associated with differential equation. In this paper, using the shift operator, a subspace G p),ρ (0, 1) is considered, in which the basis property of exponentials and trigonometric systems of sines and cosines is established when the weight function ρ(·) satisfies the Muckenhoupt condition. It is proved that the system of eigenfunctions and associated functions of the discontinuous differential operator corresponding to the given problem forms a basis in the weighted space G p),ρ (0, 1) ⊕ C,1 < p < +∞ with the weight ρ(·) satisfying the Muckenhoupt condition. The question of the defect basis property of the system of eigenfunctions and associated functions of the given problem in the weighted spaces G p),ρ (0, 1),1 < p < +∞, is considered.