Evereven or Ever-eve was a poetic name for the region of Aman from which Eärendil set out on his epic journeys beyond the World. It appears only in Bilbo Baggins's 'Song of Eärendil' at Rivendell, and so the term may have been meant as Bilbo's own invention. In the poem, it's described as being a land of hills and fountains.
In the original conception, Eärendil undertook an epic journey carrying him on a cycle from Evereven through Evernight, Evernoon and Evermorn. Much of this was lost from the final form of the song, so that while the names Evereven and Evernight both survived into The Lord of the Rings, their context and meaning have become difficult to identify.