The EGRET telescope aboard the NASA Compton Gamma-Ray Observatory (CGRO) has repeatedly detected 3EG J1835+5918, a bright and steady source of high-energy gamma-ray emission which has not yet been identified. The absence of any likely counterpart for a bright gamma-ray source located 25° off the Galactic plane initiated several attempts of deep observations at other wavelengths. We report on counterparts in X-rays on a basis of a 60-ks ROSAT HRI image. In order to conclude on the plausibility of the X-ray counterparts, we reanalysed data from EGRET at energies above 100 MeV and above 1 GeV, including data up to CGRO observation cycle 7. The gamma-ray source location represents the latest and probably the final positional assessment based on EGRET data. We especially address the question of flux and spectral variability, here discussed using the largest and most homogeneous data set available at high-energy gamma-rays for many years. The results from X-ray and gamma-ray observations were used in a follow-up optical identification campaign at the 2.2-m Guillermo Haro Telescope at Cananea, Mexico. VRI imaging has been performed at the positions of all of the X-ray counterpart candidates, and spectra were taken where applicable. The results of the multifrequency identification campaign toward this enigmatic unidentified gamma-ray source are given, especially on the one object which might be associated with the gamma-ray source 3EG J1835+5918. This object has the characteristics of an isolated neutron star and possibly of a radio-quiet pulsar.