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In Chapter 16, just before Huck paddles to shore to find someone and relieve his conscience, Jim does something that ultimately changes Huck's mind.
What does Jim do?
Jim dances so happily, Huck feels too bad to ruin it.
Jim says, "Jim won't ever forgit you, Huck; you's de bes' fren' Jim's ever had; en you's de only fren' ole Jim's got now."
Jim reveals his plan to steal back his family, which Huck thinks is noble.
It is not anything Jim said, it's the men he encounters. They refer to Jim as a "nigger" which offends Huck.
Jim does not do anything specifically; Huck's conscience just gets the better of him.