Haya was first named by Peter J. Makovicky, Brandon M. Kilbourne, Rudyard W. Sadleir and Mark A. Norell in 2011 and the type species is Haya griva. The generic name and the specific name are both derived from the Sanskrit for the Haryagriva avatar of the Hindu god Vishnu, depicted with a horse head. This name refers to the elongate horse-like skull of Haya and the appearance of this deity in the Buddhist art of Mongolia.