I see that this is from (a WONDERFUL book): "Because of Winn-Dixie," by Kate DiCamillo. To a native speaker, iIt is in fairly easy English, but it has colloquialisms and US cultural references. So keep asking questions in this forum. In this case, Miss Franny is an old woman, and she speaks somewhat old-fashioned and formal English. "Abiding" means something that persists, remains, stays, and does not change. A "notion" can mean "a silly idea" or "a mistaken idea." She says men have the idea that war is fun. "Notion" means that she thinks it is a silly, mistaken idea. She says it is an "abiding notion" means it can't be changed. To "convince" someone is to make someone accept an idea. To "convince someone differently" means to make someone change their mind. "He wanted to take Route 95, until I convinced him differently. I pointed out that there was construction going on. I convinced him that a different road would be better."

(Replying to your reply) "To convince them" means "to get them to accept a belief." "To convince them differently" means they are already convinced of something--that "war is fun." She wishes that they could be convinced of something different--that war is not fun, but that "war is hell."