In this paper, ion-selective electrodes sensitive to copper(II) ions are presented, in which new composite, synthesized from copper(II) oxide nanoparticles (CuONPs) and multi-walled carbon nanotubes (MWCNTs), was used as a solid contact. For comparison, electrodes obtained using separate components of the nanocomposite, i.e., CuONPs and MWCNTs, as well as unmodified electrodes, were also studied. The tested nanomaterials have been applied in two ways: as an intermediate layer placed between the ion-sensitive membrane and the internal electrode, and as an additional component of the ion-selective membrane mixture. To investigate the influence of the electrode’s structure modification, the selected analytical parameters obtained by potentiometric measurements (slope, linearity range, detection limit, potential stability, and reversibility) and electrochemical impedance spectroscopy measurements (membrane resistance and charge transfer resistance as well as double layer capacitance) were determined and compared. It was found that the use of all nanomaterials improves the properties of the electrodes, with the effect being the strongest for electrodes modified with the CuO-MWCNTs nanocomposite. The nanocomposite-based electrodes, both those with an intermediate layer and those with a nanocomposite-modified membrane, showed a Nernstian slope of the characteristic, a wider working range and a lower detection limit compared to unmodified electrodes. Moreover, application of all nanomaterials, especially nanocomposite resulted in improvement of both, stability and reversibility of the sensor potential. Modification of the electrodes did not make them sensitive to changing external measurement conditions (lighting, presence of gasses, redox potential). The electrode with the best parameters (based on nanocomposite) was successfully used to determine the Cu2+ ions content in tap water and mineral water, obtaining satisfactory results.