The formation of negative hydrogen ions in a conventional hollow cathode discharge has been investigated. A mixture of Ne and H-2 proved to be more advantageous compared to pure hydrogen. The study has been performed by solving the electron Boltzmann equation, coupled with a system of balance equations for neon and hydrogen neutral and charged particles. The vibrational distribution function of hydrogen has been calculated. Our calculations show unusually high population of vibrationally excited hydrogen molecules in a Ne-H-2 mixture, which explains the high density of negative hydrogen ions under optimal conditions (total gas pressure of few Torr, hydrogen number mole fraction of 1-10% and discharge current of 10100 mA). Line intensities originating from highly excited neon states vs. hydrogen pressure have been calculated and a comparison with existing experimental results has been made.