My first interpretation of “Into the West” is that it is a song that was supposedly written for someone dying, someone young and unprepared to face death.
The first time the theme of “Into the West” occurs in The Lord of the Rings is during a hopeless battle that will likely end in the eradication of the free world, and death to anyone fighting on the side of freedom. Pippin, a young soldier who expects to be killed in the fighting within the next few minutes, asks Gandalf, someone who has actually experienced death, what it will be like. Gandalf comforts Pippin, describing the afterlife as a beautiful and peaceful place. The descriptions Gandalf uses are also in the song.
At first the song seems to be a lullaby based on the first two lines
your sweet and weary head.
Night is falling.
You have come to journeys end.
As well as the repeated comforting of the recipient of the song, likely a soldier who was not expecting his life to be cut short so soon, that it is safe to rest and are bidden to “sleep.”
However, later in the song the lyrics make it quite clear that “sleep” is just a metaphor for death. This is supported by several lines in different verses.
The second verse says “dream of the ones who came before,” perhaps in reference to ancestors, “They are calling, from across a distant shore.” Meaning “those who came before” are waiting to receive the one who is dying, to welcome them into the afterlife.
The third verse is more or less a admonition for being afraid of death, that the inevitable is not necessarily bad, and an encouragement to let death come like sleep.
The first part of the chorus is the singer wondering what death is like, what death looks like, answers to unanswerable questions. It’s supposed to be wonderful. Death is a good thing. It happens to everyone.
After the chorus, the fate of the dying soldier is sealed, his life is at an end. But that is not the end of all things, relationships will be renewed after death, death only separates for a while.
The chorus again, death is a good thing, everyone passes the same way.