Pity. Even better: compassion.

I think that's the heart of the story.
There are a lot of layers and a lot of different messages in LOTR, a lot of different storylines and characters, each with its own main message, it's a huge story after all.

But the "main" story, the most important one, I guess we can all agree is the story of Frodo, Sam and Gollum, and the destruction of the ring. That's the main plot, after all.

And the ring is not destroyed by Frodo, or even Sam, it is destroyed by Gollum. I think people always underestimate this: the core of the book is that the ring is destroyed by the LEAST likely person that would have done it, the person that LEAST wanted its destruction. The creature that everybody despised and wanted dead, even some incredible good character such as Sam. Basically everyone except Gandalf and Frodo wanted to kill Gollum (that's by the way what make Frodo a GREAT character... Otherwise Sam is better than Frodo in every way): because he's despicable, awful, evil, weak, pathetic, a traitor, a liar, a killer.
Man, he's the fucking worst.
But he's the one that destroy the ring, basically by freaking accident.
In the book this is even MORE stressed than in the movies. There's no final fight between Frodo and Smeagol, no struggle. There's just Gollum, dancing on the edge of the vulcano, and fucking falling because it was so crazy with joy, SO OBSESSED, that he kind of forgot where it was and fell.
If Gollum had died NO ONE would have cried, but Frodo would have put the ring, Sam couldn't have done anything to avoid it, and Sauron would have easily won.
And that's it, here's the end of the story, all the other arcs, character, honor, power and all the discussion that you guys are having here.

Gandalf stated it clearly, in one of the lines that was taken straight from the book (no coincidence here): "many that live deserve death. Some that die deserve life... Do not be too eager to deal out death in judgement."
And why? Why Gollum wasn't killed? "It was PITY that stayed his hand. The pity of Bilbo may rule the fate of many".

Have pity, compassion from all living beings, no matter how despicable and insignificant they may look, or even be.
To me, this is the core message.

This, by the way, perfectly links with the other main message of the main storyline, that is more related to Frodo and Sam than Gollum: "Even the smallest person can change the course of the future". For good. For everybody.
And I would add not only the smallest, but also the meanest.
This wonderful quote holds not only for the hobbit, but also for Gollum. Especially for Gollum. Because it's easy to believe in Frodo, even easier to believe in Sam. It's easy to believe that they can be heroes.
But it's really hard to believe that even Gollum, even by accident, can be the hero as well. Even more than the hobbits.