What had Holmes thought was the significance of Julia's last words in "The Adventure of the Speckled Band"? In what way could her words be...

Julia Stoner's last words involved a speckled band, which might lead people to think a gypsy did it.

Helen Stoner comes to hire Sherlock Holmes because her twin sister died in suspicious circumstances, and she is afraid she is next.  She shares the story of her wicked stepfather and his aggressive behavior, and tells Holmes about Helen’s last words.

At first I thought that she had not recognised me, but as I bent over her...

Julia Stoner's last words involved a speckled band, which might lead people to think a gypsy did it.

Helen Stoner comes to hire Sherlock Holmes because her twin sister died in suspicious circumstances, and she is afraid she is next.  She shares the story of her wicked stepfather and his aggressive behavior, and tells Holmes about Helen’s last words.

At first I thought that she had not recognised me, but as I bent over her she suddenly shrieked out in a voice which I shall never forget, ‘Oh, my God! Helen! It was the band! The speckled band!’ 

Holmes is more interested in the whistle and the metallic sound than the "band."  It is what makes him think that it was definitely murder.  The door was secured, so no one knows how anyone could have gotten in to murder her.

Holmes asks what killed Julia, asking Helen specifically about poisons.  Then he mentions the band in Julia’s last words.

“Were there gipsies in the plantation at the time?”

“Yes, there are nearly always some there.”

“Ah, and what did you gather from this allusion to a band—a speckled band?”

This is red herring.  At the time, we assume that the speckled band is the “band” of gypsies.  It turns out to be a snake.  The whistle was Dr. Roylott’s way of calling it.  When Julia told Helen about the “speckled band” with her last words, she was not talking about a group of people at all.  She was talking about a snake!

Holmes and Watson go to Stoke Moran and test Holmes’s theory.  They hear a whistle, and in comes a snake Holmes calls a swamp adder.  It is one of Dr. Roylott’s many exotic creatures, and also his murder weapon.  The snake turns on him, killing him as well.  Case solved!  No trial is needed for this one.