Eternity graffiti, Arthur Stace
Eternity is an intriguing option as a girls baby name. While word names are always popular, those with grand meanings are favorites. Serenity, for example, has been climbing the charts steadily and now sits high at #71. Eternity, however, remains rare. In 2015 there were only 84 baby girls given the name, which is about the same amount for the past decade. It has only been used since about 1990, which is when Calvin Klein introduced the perfume Eternity. However, the universal concept of eternity (mathematical, philosophical, religious or otherwise) has been around for, pun intended, eternity, and there is no off-putting association for it.
From old Latin aeturnus, Eternity means "without beginning or end." Synonym: forever. It is easy to see as a name - just like Destiny or Trinity. It has a modern, feminine sound, like Eve, Lily (currently ranking at #25) or Emily (currently ranking at #8). Nicknames could be Etty, Teri, or Terny, but I'm sure you could get creative - but it doesn't need a nickname.