'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.
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.