The phrase "My Ántonia" is an interesting one and comes up more than once in the course of the text. We see Jim-the-narrator choose this as a title for his memoir during the introduction. He starts by titling it "Ántonia," then changes his mind and writes "My Ántonia" instead. We hear Mr. Shimerda call his daughter "my Ántonia," and Jim refers to her this way several times in the novel.
The term isn't so much about ownership as it is about endearment. Jim (and the others who use the phrase, like Mr. Shimerda) feels that Ántonia is a part of his life and he a part of hers. As he tells her at the end of the novel, "I'd have liked to have you for a sweetheart, or a wife, or my mother or my sister – anything that a woman can be to a man. The idea of you is a part of my mind. You influence my likes and dislikes, all my tastes, hundreds of times when I don't realize it. You really are a part of me" (4.4.7).
The title also lets us interpret Jim's memoir as a sort of tribute to Ántonia. Through his memoir, he pays tribute to Ántonia and to the memories they shared together. It reminds us that he's not writing the memoir of his childhood, but the memoir of his childhood with Ántonia. The title centers the story and its otherwise scattered, episodic plotline around a single focal point: Ántonia.