Before we get into the deep meaning of the title Persuasion, a useful fact: Austen didn't actually call the book that. As far as we know, she meant to use the title The Elliots, but she died before the book was published, and her family decided to go with the catchier Persuasion. That bit of historical trivia doesn't mean the title is completely meaningless, however: after, all, Austen's heirs didn't just pick the title out of thin air. Forms of the word "persuade" pop up all over the text, and forms of persuasion – which can mean both "to convince someone of something" and "to believe something to be true" – again and again shape the development of the story.
From the beginning, the novel is founded on the fallout from a major act of persuasion. When Lady Russell persuades Anne not to marry Captain Wentworth eight years before the novel begins, she thinks she is doing the right thing by convincing Anne that it's her moral duty not to make herself dependent on a man who can't support her. When a sadder, wiser Anne looks back on the moment which has haunted her ever since, she doesn't blame Lady Russell for being persuasive, nor herself for giving into persuasion, but still thinks that if the power of persuading had been in her hands at that moment, she and Wentworth would already have several fat babies to their credit.
Reading that might make you think that the lesson from Anne's experience is obvious: when other people try to persuade you, it's time to put your fingers in your ears and sing "la la la, I can't hear you" until they go away. But Persuasion's stance on persuasion isn't that simple. Louisa's refusal to listen to others lands her with a head injury and a personality transplant, causing Anne to hope that Wentworth learns that "a persuadable temper might sometimes be as much as favor of happiness, as a very resolute character" (12.71). In Persuasion, persuasion in itself is neither always good nor always bad: it depends on how you use it.