The editors at the Euskaltzaindia (Basque Language Academy) website say that the name Eneko may have come from the Basque word ene, meaning "my", combined with -ko, a diminutive suffix. [noted -ed]
'Eneko' it's said to come from 'oneka' meaning 'the best'. It also was the name of Navarre's first king, Eneko Aritza, translated to Spanish as Íñigo Arista.