There is an intimation of reflection in the title. The "remains of the day" helps to imply how Stevens reflects on his life at this particular point. So much of it has passed so that there are only "the remains." Stevens analyzes these remains of his life, to see what has been passed up and what has been acted upon. In reflection about his own life, one sees only the remains of what has transpired. In its wake, Stevens sees the pain of misplaced loyalty, lost opportunities, and the denial of personal growth.
At the same time, the title might also refer to how there is something left upon which to build. Darlington Hall being taken over by Mr. Farraday represents a change. There are some "remains" left of the day in which life can be lived with more purpose and more relevance. The title works because it operates on such a level where it is both tragic and life- affirming elements simultaneously. While there is much in way of pain as to what has passed, there is also a clear understanding that what does remain can be lived with meaning. Stevens is at a point where reflection about the missed chances in his own life can translate into action for the present and the future with a sense of understanding.