"Coming of age" basically just means growing up—and in The Breakfast Club, the main characters have to do this while dealing with a world of clueless adults who don't understand anything and are frequently hostile.
This process involves a change in perception more than anything—they need to learn how to perceive each other and their struggles in a more enlightened way. Yet, even though they need to "come of age," they're trying to avoid becoming like Richard Vernon and other adults who never make this leap in perception and still view people according to their biases and preconceptions.
John Steinbeck wrote, "When a child first catches adults out—when it first walks into his grave little head that adults do not always have divine intelligence, that their judgments are not always wise, their thinking true, their sentences just—his world falls into panic desolation." You could argue that this describes the attitudes of the five main characters in The Breakfast Club.
In the 1960s a popular slogan was, "Don't trust anyone over thirty." The Breakfast Club seems to hint that this is good advice, implicitly arguing that younger people are truer of heart than older people.
No one in The Breakfast Club is ultimately all that happy with what they've got, whether what they have would typically be judged "good" or not. What John Bender has is pretty terrible by any standards: an abusive dad and a lousy home life. But what Claire Standish or Andrew Clark have looks a lot better, since they're swimming in popularity and social approval. Yet they're totally dissatisfied too. It's as though dissatisfaction were the natural atmosphere of life itself.
Sigmund Freud once wrote to a patient, "But you will see for yourself that much has been gained if we succeed in turning your hysterical misery into common unhappiness." But can the characters ultimately surmount either their "hysterical misery" or the "common unhappiness" that is a part of life?
You could argue that real satisfaction and real joy are actually obtainable in life. In The Breakfast Club, the understanding they gain with each other might actually prove key to this attainment, and the happiness of being with people really does amount to something lasting.
No one gets along with their parents in The Breakfast Club. Bender's father abuses him, even burning him with a cigar at one point. The other kids' parents aren't physically abusive, but they don't understand their kids and try to force them into things they'd rather avoid. It's a pretty bleak depiction of family life. If you're someone like Allison Reynolds, you're pretty much alone in life, unless a bunch of friends suddenly fall into your lap (as happens in the movie).
The great writer Tolstoy once claimed that, "All happy families are alike. All unhappy families are unhappy in a different way." You could say this is true of The Breakfast Club: Brian, Andrew, Allison, Bender, and Claire all experience distinct kinds of suffering through family life.
On the other hand, you could argue that there are huge similarities in the way family life makes them unhappy. Brian and Andrew both suffer from different kinds of pressure—academic and athletic—while Allison and Bender both deal with apparent neglect. And Claire shares something that they all have: Their parents don't treat them like people, but as a means to an end.
Before the action of The Breakfast Club starts, the characters' friends are all drawn from their own cliques. Those friendships are based on shared stereotypes (they all identify as jocks or criminals or rich kids or whatever), but not really on shared experiences.
By the end of the movie, we see that the different characters all really have a lot in common with each other and aren't nearly as divided as they thought. So the basis of friendship switches: Instead of having society assign you an identity and friends along with it, you discover your own shared identity with a diverse group of people and are suddenly free to make friends with anybody.
Samuel Johnson said that most friendships are mainly "confederacies of vice and leagues of folly." You could argue that this is true of The Breakfast Club: They smoke pot together, and Claire gets with Bender just to anger her parents.
Or, you could argue that friendship is or can be way deeper than that. In The Breakfast Club, friendship teaches the characters to see themselves more clearly by seeing other people more clearly. In order to grow in self-knowledge, they need to grow in their knowledge of other people.
This might be the big theme of The Breakfast Club, the one that knits all the other themes together. Everyone in this movie has an identity socially defined by their peers and by the adults who are supposed to know better.
Brian reads them off at the beginning of the movie: "a brain, an athlete, a basket case, a princess, and a criminal." Everything that happens in the movie involves learning that each of them has more to offer than these glib descriptions would indicate. They're all deeper, fully formed people—dynamic characters, not stock characters.
Our identity is something determined by society. You can see this in the way the characters all act according to social roles they've been assigned—nerd, jock, etc.
Our identity is something deeper than what society determines it is, and we have some sort of transcendent inner core. You can see this in the way the characters are able to overcome their social roles and ultimately relate to each other in a genuine form.