The Eight Melodies. No matter how it's covered, be it 8-bit, rock, or orchestral, it expertly conveys the sense of childhood adventure. It subtly invokes fear, innocence, heartbreak, and hope, all at once. There's a reason so many professional covers exist.
The idea of the Eight Melodies in the game in general. Think about it; A childhood antique doll that used to be played with many times, a mother and a child bird separated by humans who didn't know any better, a monkey who's caged and locked up in a zoo, a dusty old piano that plays on its own in a long abandoned building, a lone cactus in an empty desert where a war happened and exterminated the rest of its friends, a dragon who's been sleeping away its life, a robot companion who defends you from everything, and lastly, the tombstone of your dead and loving great-grandfather who built the robot companion knowing that you might need it someday.
Or if you're playing the original Famicom version, the eighth melody comes from Queen Mary herself, who turns out to be your great-grandmother, Maria. The woman who spent her life trying to remember the Eight Melodies dies in peace when she can complete the song and sing it on her own.
Try reading the above in one go without getting tears in your eyes.
Queen Mary's final scene. Her story in general is tearjerking.
"Oh! Giygas! I loved him... I loved him, as if he was my own child... He was always wagging his tail... just like a pup..."
The entire sequence with Magicant disappearing is this. You've quickly grown to think of it as a safe haven you can return to with the Onyx Hook at any time. It's where you can reliably grind the party members up to where they can effectively fight for themselves. It's an easy place to get a free heal should you be in desperate need of one, and some of the best equipment in the game is located there. Then after Queen Mary remembers the eight melodies and her own past, you watch as it all deteriorates and fades away. The haunting rendition of the eight melodies playing in the background the entire time it happens has to make it one of the most emotionally charged cutscenes of any NES game created...
Teddy's fate at the hands of R7038. He gets better. (At least in the English prototype and the GBA remake.)
Speaking of which, the song that plays at his death bed is a heart wrenching remix of Pollyanna. When you consider the lyrics and Teddy's supposed final words, it's easy to how some view it as the saddest part of the game.
EVE's death. The game itself doesn't make much of it, but most of the fans take it really hard. Just think about it: EVE was built by George, Ninten's late great-grandfather, specifically to protect Ninten and his friends, which she manages to do... at the cost of her own life.
During the quest to find Pippi, there is a small implication that Pippi's father died and was buried in the cemetery south of Podunk, and Pippi went there presumably to see him again...
The Flying Men are NPC party members that Ninten can recruit on his journey by visiting a house in Magicant and talking to one. However, there's a catch. Unlike regular party members, if they run out of HP, they are dead for good. The kicker? A gravestone is erected for the fallen Flying Man next to their home. The worst part is that there's only a finite amount of them in every playthrough, so killing them all is a very real possibility.
"To Ninten's horror and dismay, the brave soldier, Flying Man, rests in peace here."