In addition to the examples given by the previous posts, Susan E. Hinton's The Outsiders continures to be pertinent in the 21st century in part because of its gang-related themes. Teen gangs were still in vogue when the story takes place in the mid-1960s, but gang activity declined drastically for several decades afterward. With the rise of gangs during the 1990s forward, present day students can once again see a correlation between greasers, Socs and the far more violent gangs of today.
I would say that the reason why the book is read today because Hinton struck a chord of adolescence that resonates with all youth. The idea of being displaced from the social setting, dislodged or incapable of fully immersing oneself in their family settings, and finding solidarity with other teens are themes that seem to be universally appealing to all adolescents. The moments in which the characters must define themselves and their identities, align themselves with what they believe is right, and constantly battle others' perceptions of them make the work something to strike connection with young readers. While some of the jargon and settings might be locked into the time period, I think the overall universality of the characters, their motivations and predicaments, and the nature of how social orders are structured will always speak loudly to adolescents and allow the book be read by future generations of both "Socs" and "Greasers."
Everybody loves a good rumble! The Outsiders features interesting characters: beer-on-cereal-instead-of-milk kind of characters; as well as sympathetic characters: kid-saves-kids-in-fire-and-dies kind of characters. By the end of the novel readers even feel sorry for the real jerks in the novel, like Dallas.
The novel also makes at least a slight attempt at presenting both sides of a difficult story. The narrative doesn't spend much time on the Socs, but it does spend a little.
Finally, the novel is a simple and emotional read for younger readers. It won't be read forever by older readers, but it will probably be around for a long time with middle schoolers.
In my opinion, this book will still be read for a long time because the issues it addresses are just about timeless (at least, they are very common to teens in the Western world over the last sixty years or so).
The book explores the need that teens feel to be accepted by someone. Teens are at a stage in their lives where many of them feel that they are not sure of who they are or how they fit into society. They often feel that much of the world is against them.
This book looks at these feelings. Because of that, it makes as much sense to read it now as it did when it first came out.