Spinner's name means "someone/something who spins". In the original Japanese, his name is just the romanization of the English word 'spinner' - 「スピナー」 (supinaa).
When characters in the manga define it, they often use 「紡ぐ者」 (tsumugumono) lit. 'a person who spins'. 「紡」 is the key character here, meaning spin, in the way one spins yarn, or spins a story.
Spinner deliberately choose this name because he wanted to 'spin' Stain's dream into reality.
Later, when he abandons following Stain to follow Shigaraki, All For One takes notes of his meaning of his name, to tell him that he'll be helping Shigaraki to 'spin' his goal [into reality]. (Viz translates this as "support Shigaraki Tomura in his crusade and do justice to your name... as one who spins this tale.")
This is appropriate, because during and since the MVA arc, Spinner has been doing everything he can to support Shigaraki's dream of destruction - to achieve their goal of 'that beautiful horizon'. Throughout the third act, Spinner's still trying to spin that dream into being.
And it is their goal - it was Shigaraki who first talked about the 'horizon' in his speech to Ujiko, the speech that affected Spinner so much he started down the path of devotion to Shigaraki; so that Spinner would be the only one out of everyone in the League to see Shigaraki's horizon in Deika (the prettiest thing he has ever seen). Somehow, Shigaraki figures this out, so that much later, when Shigaraki is preparing to decay Mt. Fuji, he dedicates this destruction to Spinner - to "build the horizon... that Spinner's been looking forward to."
But-- Shigaraki fails. Deku stops him, and Shigaraki seemingly dies. He dies without having built that horizon, without having destroying anything.
In Shigaraki's final moments, Deku tells him, "I wanted to stop you. I wanted you to stop yourself. To keep that grief and misery from spreading any further."
In Japanese, when Deku talks about this 'cycle of sadness', he says he wants it to 'stop spinning' - 「紡がれない」 (tsumugarenai). His line uses the same character meaning 'spin' as the one I talked about above - 「紡」
lit. 'so the sadness won't spin'
Shigaraki listens to Deku's words, and after a moment, says to him:
"If Spinner is alive... tell him Shigaraki Tomura fought to destroy to the bitter end."
I thought before and still think it's Shigaraki leaving some words of comfort for Spinner. He failed to destroy everything, failed to succeed in reaching his goal (which is Spinner's goal, because it's Shigaraki's goal, because it's the goal Shigaraki made for the League); but he did try his hardest. He died trying to achieve their horizon, because until the very end, he was keeping their promise as best as he could.
Maybe, it's also: don't worry about the failure; all the way until the end of his life, Shigaraki Tomura got to chase after his dream, their dream.
Maybe even, when taking all of the context from above and putting it into these last words of Shigaraki's: but it's stopped. Thus, Spinner doesn't have to spin for him anymore. The sadness has stopped spinning. Maybe: If Spinner stops on his own, Heroes won't have to stop him. If he's still alive, he can stay alive.
And see, the Shigaraki that says these words is the 'same' Shigaraki as the one in Spinner's memories of the two of them talking about games together. You can tell by the visuals:
In the panel right before Shigaraki tells Deku to deliver a message, the lock of hair on Shigaraki's face falls below his nose; and the locks of hair that frames his face falls below his chin.
But in the panel where he gives those very last words, he's wearing a black shirt. His lock of hair on his face does not reach past his nose. The locks of hair framing his face ends at the level of his mouth.
So this is the Shigaraki from Spinner's memories of them being just two gamers. This is the moment they weren't Villains or boss and subordinate; they were just two guys, close in age, (getting along better than Spinner thought, bonding over games and stuff), being friends.
Shigaraki is giving his last words as Spinner's friend; and they are to tell him, i kept our promise. i chased our dream.
