Identifying active galactic nuclei (AGNs) through their X-ray emission is efficient, but necessarily biased against X-ray-faint objects. We aim to characterize this bias by comparing X-ray-selected AGNs to the ones identified through optical variability and mid-infrared (mid-IR) colours. We present a catalogue of AGNs selected through optical variability using all publicly available z-band Hubble Space Telescope images in the GOODS-South field. For all objects in the catalogue, we compute X-ray upper limits or discuss detections in the deepest available ~7 Ms Chandra Deep Field South images and present the Spitzer/Infrared Array Camera (IRAC) mid-IR colours. For the variability study, we consider only sources observed over at least five epochs and over a time baseline of up to 10yr. We adopt the elevated median absolute deviation as a variability indicator robust against individual outlier measurements and identify 113 variability-selected AGN candidates. Among these, 26 have an X-ray counterpart and lie within the conventional AGN area in the FX/Fopt diagram. The candidates with X-ray upper limits are on average optically fainter, have higher redshifts compared to the X-ray-detected ones and are consistent with low-luminosity AGNs. Out of 41 variable optical sources with IR detections, 13 fulfill the IR AGN colour selection criteria. Our work emphasizes the importance of optical variability surveys for constructing complete samples of AGNs including the ones that remain undetected even by the deepest X-ray and IR surveys.