#Harper's Dictionary of Classical Literature and Antiquities - Google Books

[thinksandthings]

Elysium is a lovely sounding word which in English means "a place or state of ideal happiness," but its primary original definition comes lifted straight from Greek mythology, where Elysium was the name of the home of the blessed souls after death. This word in Ancient Greek was Hλύσιον, Ēlúsion. This same mythical location was also referred to as the Isles of the Blessed.

In older writings, it was specifically reserved for those singled out and favored by the gods. Heroes and other immortals would be rewarded with a place here to pass the rest of their eternity in bliss. As time went on, the concepts of punishment and reward in the afterlife became more widespread, which eventually merged Elysium more deeply into the concept of afterlife for all people. It became one of several possible levels of Hades which departed souls might find themselves in, depending on a favorable judgement passed on their conduct by the Underworld judges Minos, Rhadamanthys and Aeacus.

Importantly though, Ancient Greek religion was hugely varied and constantly evolving, so this wasn't the only theory or iteration.

There doesn't seem to be any agreement on the etymology of the word beyond the Ancient Greek, but one source I found suggested that it might have something to do with another Greek word, which is the verb meaning "to relieve or release." In that interpretation, Elysium might have been the paradise which promised relief from the pain and burdens of living.