The name of the patriarch, Հայկ Hayk, is not exactly homophonous with the name for "Armenia", Հայք Hayk‘. Հայք is the nominative plural in Classical Armenian of Հայ (Hay), the Armenian term for "Armenian." The etymology of Hayk' (Հայք) from Hayk (Հայկ) is impossible. The origin of the term Hay ("Armenian") is obscure. Nevertheless, both Hayk and Hayk' are usually connected to Hay (Հայ) and Hayer (Հայեր, the nominative plural in Modern Armenian) -- the self-designation of the Armenians.
Hayk would then be an aitiological founding figure, like e.g. Dan for the Danes, Seaxnēat for the Saxons, etc. The standard etymology for hay-, the self designation of the Armenians, is a derivation from Indo-European *poti-s, meaning "lords". These tribes may have settled Armenia from the Mitanni kingdom, when Sargon II mentions a king of part of Armenia who bore the Indo-Iranian name Bagadatti ("Theodore"). A connection made in Armenian historiography of the Soviet era, with Hayasa mentioned in Hittite inscriptions, is not tenable on phonological grounds. The Armenian word haykakan (Armenian: հայկական, meaning "that which pertains to Armenians") finds its stem in this progenitor.