The origin of Elysium
The lyrics of the famous "Ode to Joy" (by Schiller, on the final of Beethoven's 9th) mention the 'daughter of Elysium'. The term Elysium or 'Elysian plane' go back to Hesiod, Homer and Aeneid.
The definition seems to be either paradise or afterlife, perhaps both.
What's the origin of Elysium and how should it be translated in different contexts?
According to the comprehensive Theoi.com page Realm of Elysion
THE ELYSIAN FIELDS was the final resting place for the souls of heroes and virtuous men. The ancients often distinguished between two such realms--the islands of the Blessed and the Lethean fields of Haides.
The two realms are for heroes and the virtuous, described as:
"White Island" or "The Islands of the Blessed", was an afterlife realm reserved for the heroes of myth.
A netherworld realm that is located in the depths of Haides (beyond the river Lethe). A realm promised to "initiates of the Mysteries" who had lived a virtuous life.
However, according to the website, Elysium was and still is an 'evolving concept', where
Homer knows of no such realm, and consigns all of his heroes to the common house of Haides, while Hesiod and many other poets speak only of a paradisal realm reserved for heroes. Roman writers (such as Virgil) combine the two Elysia--the realm of the virtuous dead and the realm of heroes become one and the same.
An important note is that the term Elysium (and Haides) were not proper nouns in Ancient Greek, but rather as an adjective - the website provides examples:
pedion Elysion (the Elysian plain) and domos Haidou (the domain or house of Haides). The etymology of elysion is unclear. It may be connected with the Greek verb eleusô (eleuthô), "to relieve" or "release" (i.e. from pain), and/or with the town Eleusis, site of the celebrated Eleusinian Mysteries.
An interesting description about the term "Daughter of Elysium" from the classic Ode to Joy from the Wikipedia article Elysium: Post classical literature, is that:
After the Renaissance, an even cheerier Elysium evolved for some poets. Sometimes it is imagined as a place where heroes have continued their interests from their lives. Others suppose it is a location filled with feasting, sport, song; Joy is the "daughter of Elysium" in Friedrich Schiller's ode "To Joy"
Particularly when the verse in its entirety is considered:
Joy, thou beauteous godly lighting,
Daughter of Elysium,
Fire drunken we are ent'ring
Heavenly, thy holy home!
Source: The Schiller Institute
Somewhat anecdotally related, in the past (many moons ago), I played Ode to Joy in an orchestra, when it gets into the full swing, one can imagine that they are among the heroes and virtuous of old in an environment that is joyous, brave and virtuous.