Melancholy Moon

Cold hearted orb that rules the night
Removes the colours from our sight
Red is grey and yellow, white
But we decide which is right
And which is an illusion

—The Moody Blues

The Moon. Cold and distant, alone and lifeless... yet, it's the only celestial body whose light can pierce the pitch black night. So when a character is shown moon gazing, it means he or she shares many of these lunar traits: they are isolated and introspective, melancholy... yet oddly hopeful. All this just by having them look quietly at the moon, or using it as a big old backdrop.

He or she may have a Tragic Dream, have suffered a Dark and Troubled Past, or have embarked on some new quest that promises to change their life. The trope is also frequently used in love songs.

Can overlap with Weird Moon, often to emphasize the moon. In this case, the moment is used both to establish the world and the character.

Contrast Watching the Sunset and Cue the Sun.

Examples of Melancholy Moon include:

Anime and Manga

In Neon Genesis Evangelion just before Shinji and Rei prepare to go snipe the angel, during which they muse about why they pilot the Evangelions, the binds they have with people, and Rei promises Shinji that he won't die because she'll protect him, ending the scene with "Sayonara" with a big full moon over her shoulder. The whole scene was very melancholy and introspective.
One of the recurring songs in BECK is "Moon on the Water". The song is in of itself a melancholic love song but in episode 5 when both Koyuki and Maho skinny dip in the school pool they sing it song together under a summer full moon. You can't get more romantic than that.
In Darker than Black, the Emotionless Girl and White-Haired Pretty Girl Yin gets a lot of symbolic focus on the moon in her flashbacks to her tragic childhood, though the moon is admittedly absent from the sky in the present. There are connections beyond her melancholy demeanor, since the moon is traditionally associated with mediums, and for a bonus, "Yin" is Chinese for "silver". And late in the second season, the moon returns, bloodred, in circumstances related to an Eldritch Abomination possessing Yin.

Film

In Field of Dreams.
In Harry Potter and the Philosopher's Stone, after he arrives in Hogwarts and everyone is in bed, Harry's up looking out his window at the moon. It fits into the "melancholy yet hopeful" category as it marks the beginning of a new life for him, yet it's a somewhat sad scene because of the backstory with his parents.
Smeagol fishing in the waterfall in The Lord of the Rings: The Two Towers has a curious use of the moon. Smeagol, who has found a measure of redemption helping Frodo and even banishing Gollum, is "betrayed" by Frodo under Farmir's threat of killing Smeagol.
Both the big baby and the clown in Toy Story 3 do this. Which is pretty darn appropriate considering their shared backstory.
In An American Tail, both Fievel and Tanya stare at the rising moon during their Distant Duet.
- Same with Odette and Derek in The Swan Princess.
The first Shrek movie has a scene in which Shrek and Donkey stare at the night sky, as Shrek talks about ogres of the past being among the constellations. It ends with them looking at the moon, accompained by this exchange:

Donkey: So, uh, are there any donkeys up there?
Shrek: Well, there's, um, Gabby, the Small and Annoying.
Donkey: Okay, okay, I see it now. The big shiny one, right there. That one there?
Shrek: That's the moon.

Mulan, during the "I'll Make a Man Out of You" song sequence.
The Nightmare Before Christmas: "Jack's Lament" is built mostly around the backdrop of Spiral Hill against the moon.

Literature

Very popular in Ancient Chinese Literature, especially poems and "lyrics" (which were originally set to songs that have long been lost). Many a poem is of the poet being unable to sleep, climbing to the top of a pavillion (optional), staring at the moon and musing about their sad fate (mostly being demoted and Reassigned to Antarctica) and/or their longing for home. These were incidentally the two most commonly expressed sentiments in poems throughout the dynasties.

Music

The premise of the Rodgers and Hart song "Blue Moon," at least the first half.
Shivaree's "Goodnight, Moon."
Conway Twitty's "I Don't Know A Thing About Love" features a "Man in the Moon" who's just as melancholy as the observer, insisting he doesn't really have any answers to anyone's questions.
In the same vein, Collin Raye's "Somebody Else's Moon" is a tearjerker about a man watching the moon and thinking of it as belonging to somebody else now just as his former sweetheart does.
Beethoven's "Moonlight Sonata" is in minor and is a soulful and introspective song he wrote for a lover.
The Moody Blues, "Late Lament" on their Days of Future Passed album, quoted above.

Theatre

Giacomo Puccini, the composer of famous operas La Boheme, Tosca and Turandot, loved this trope. Almost all of his operas have the moment of Melancholy Moon, usually in connection with the main soprano heroine. In Turandot, the trope is played with in a most gruesome way: the moonrise is the signal for starting an execution. It is also averted, when Prince Calaf has his moment of moon-gazing: he is not melancholic but self-assured and hopeful.

Video Games

In Tsukihime, Shiki stares at the moon a lot when he is feeling down. As you could expect from a game whose title translates as "Moon Princess" from Japanese. And its subtitle "Blue Blue Glass Moon, Under The Crimson Air".
Lunar has a funny variation on this. Throughout the series, there's a celestial body that unquestionably fills the same role as the moon. It's distant, cold, and lifeless, but nevertheless always shining through the night and a source of hope and wonder. It's present in every night scene, most memorably shining behind Luna during Wind's Nocturne, as well as playing a massive role in the plot of Eternal Blue and being the focus of the Eternal Blue Theme. So, what's the twist? It's not a moon at all. The characters are living on its moon, called Lunar or the Silver Star, while gazing up towards the frozen planet it orbits, the Blue Star.
Fragile Dreams: Farewell Ruins of the Moon, the game in the trope picture at the top. It's used as a prominent symbol of isolation and loneliness. To really hammer the latter aspect, the girl pictured above and the male protagonist of the game may be the only two people left in the world.
The Outer Wall in Cave Story. The moon dominates the night sky, and the pensive "Moonsong" is the BGM.
Asama Sakuya from Girls Love Visual Novel Akai Ito gazes at the moon sometimes. She's the last of her kind, a tribe which had strong connection to the moon. She has been alone for the past 1700-years. In one ending, she dies, and Kei prosaically laments how Sakuya's soul is going to the moon.
One of the various NPCs on Windfall Island in The Legend of Zelda the Wind Waker spends their nights gazing up at the moon. They also whine about how nobody can understand them. One of the sidequests involves taking a picture of a "perfectly round, pale object" as a test to see if Link truly understands them.