The first clue to understanding this book's title comes when we realize that the "dove" in this story is Milly Theale. Kate bluntly says, "You're a dove" (1.5.6.23), when she is explaining to Milly why people treat her so well. Milly is delicate, innocent, and beautiful. But the image of "the dove" becomes more complicated as the book unfolds.
For Milly, a dove might be beautiful, but it's also something that is helpless when it's put in a cage and displayed for its beauty by humans. At this moment, Milly realizes that in her own life, older women like Maud Lowder and Susan Stringham try to control her and parade her around London like a trophy.
As she realizes at one point: "That was what was the matter with her. She was a dove. Oh, wasn't she?—it echoed within her." (1.5.6.23). Milly doesn't want to be a dove. Under the guidance of Sir Luke Strett, she resolves to live the rest of her life on her terms and no one else's. She tries to shed her symbolic role as a dove.
Ultimately, though, Milly can never get people to stop thinking of her as a delicate little birdie. When Aunt Maud delivers the news of her death to Merton, for example, she reports that, "Our dear dove then […] has folded her wonderful wings." (2.10.3.9). This comment shows that no matter how hard Milly tried to strike out on her own, she was seen to the end as a passive, fragile young lady and an innocent victim of life.
But Aunt Maud follows this statement with the phrase, "[u]nless it's more true […] that she has spread them the wider." (2.10.3.10). Maud means that it's possible to understand death as either a defeat or a victorious freedom. Milly might not have lived on her own terms, but she definitely died on her own terms.
So yeah, James's title is pretty dense. How Jamesian of him.