According to some sources, the novel Persuasion was not titled by Jane Austen but by her editors, since she died before titling the story. However, one can see why the title is suitable. The power of convincing, or persuading people is quite expansive in that the choices one makes as a result affect a lot of people. Sometimes the choices made after being persuaded are only in the interest of one party involved, and someone will always come out losing out.
First, it was persuasion what determined the fate of Anne when she was persuaded to quit her courtship with Cpt. Wentworth for his lack of money. As a result, she spent years as a spinster and regretted much having being persuaded.
Equally, her friend Mrs. Smith was persuaded by Anne's own father to indulge in their riches to live a pretty much "dandy" lifestyle, and due to this persuasion, the Smiths became almost bankrupt as well as Anne's father Sir Walter Elliot.
In all, one could say that Austen is sending out a message of warning for all the possible moments one will face and people will try to convince us of doing something that they think may be for our own good, when in fact it is for their own benefit. In the case of Anne, she would have made her family, which holds a Baronet, quite embarrassed if she had married Wentworth because he was poor. It was not so much for the good of Anne, but for the appearances of the family that they suggested this to her. For that reason, persuasion is a double edge weapon and needs to be treated carefully.