This story plays on the perceived pragmatism of Americans, who were, in the late 19th century, using their newly growing wealth and power to buy their way into English life. Their pragmatism collided with English culture, perceived as steeped in history and tradition.
The Canterville ghost represents the history of ancient Canterville Hall. The Otises, quite American and practical, don't believe in him. He leaves a bloodstain on the library floor. The bloodstain is inconvenient, so they simply scrub it out. It returns, which raises doubts in their minds (maybe there is a ghost), but it is not until they actually see the ghost that they are convinced it is real. Even so, the ghost doesn't frighten them.
The ghost, determined to do his job of scaring the family, starts changing the color of the bloodstain. Sometimes it is more vermilion, sometimes more purple, and once emerald green. The family treats this as a great joke--except for Virginia. The changing colors distress Virginia and she almost cries the day it is emerald green.
Through the changing colors of the bloodstains, Virginia begins to intuit that there is more to the ghost than a mere joke. Perhaps she sees the different colors as reflecting the ghost's changing emotions or as a sign of his desperation. Whatever the case, she refuses to join the family in laughing at him.
The changing bloodstains thus begin to change Virginia's heart, and when she encounters the ghost, she treats him with compassion, launching the more sentimental story of love and compassion that lurks beneath the comic tale of a ghost unable to frighten a family. Virginia shows the human side of American pragmatism: by treating the ghost with kindness, she moves Canterville Hall to a new place, a place no longer haunted by the demons of the past.
I do not think that it is the changing color of the blood stain that begins to change the Otis family's belief in the ghost. I believe it is simply the blood stain itself that causes a change in their beliefs.
When the story begins, the Otis family does not believe in the Canterville Ghost at all. They think it is completely fabricated.
"His body has never been discovered, but his guilty spirit still haunts the Chase. The blood-stain has been much admired by tourists and others, and cannot be removed."
"That is all nonsense," cried Washington Otis.
Immediately after that line of text, Washington Otis scrubs out the entire blood stain with "Pinkerton's Champion Stain Remover and Paragon Detergent." Unfortunately, the blood stain reappears the next morning. It appears again after two more scrubs as well. At that point, doubt begins to creep into each of the Otis family members.
"Mr. Otis began to suspect that he had been too dogmatic in his denial of the existence of ghosts."
During chapter 2, Mr. Otis comes face to face with the ghost himself, and all doubt is removed completely. The best part of the Otis family is that none of them are scared of the ghost. In fact, the Otis family antagonizes the ghost more than the ghost antagonizes them.
The next time that the blood stain is mentioned is at the start of chapter 3. The family has ceased to worry about the blood stain and actually finds it fun to watch it reappear in a variety of different colors.
"The chameleon-like colour, also, of the stain excited a good deal of comment. Some mornings it was a dull (almost Indian) red, then it would be vermilion, then a rich purple, and once when they came down for family prayers, according to the simple rites of the Free American Reformed Episcopalian Church, they found it a bright emerald-green. These kaleidoscopic changes naturally amused the party very much, and bets on the subject were freely made every evening."
The color change doesn't affect their belief. Their belief has already been changed. The color change of the stain is only another source of fun for the Otis family.