Conflict of Dragons from Breath of Fire III. Anyone who got to this part usually felt their heart tugging just a bit because of how well this reunion went.
Not a traditional Boss fight, and technically this theme plays elsewhere, so it's more like the Aerith scenarion but still a powerful scene if you go all the way to the end of Suikoden II.
The grand finale of the Bloodborne DLC. In truth, I never realized how peaceful and full of melancholy this track is (well the first half anyway before Phase 2 kicks in) but that might be due to not being able to hear it over the sounds of the boss screaming at you why you are screaming to your self "OH smurf MUFFINS! I'M BEING RAPED WITH AN ELDRITCH PLACENTA!!!" It's that kind of a game.
Another from Bloodborne, but from the main game. This is definitely the theme of a boss who has seen things and considering the lore of the game, this fight is more of a mercy kill than anything... if the boss would let you get an opening.
Another Aerith theme scenario, but it actually does play during the two times you fight him. Though whether this counts as an actual "sad" song is up for debate.
Final Fantasy has Aerith's Theme during Jenova LIFE, but that's not actual boss music. Same for Roxas' Theme, which was the second example I thought of after Prisoners of Fate.
I'm not sure if I'd call these sad overall, but they have sad parts, overall they still get me in the feels and they're the closest thing I could think of
First, Atelier Ayesha's final boss theme, MARIA (SPOILER)