In The Glass Menagerie, the title symbolizes Laura, the crippled daughter around whom the play develops. The different animals in her collection express the various aspects of her personality, and the ways in which they catch the light reveal Laura's changing moods.
At a more factual level, Laura can connect to her glass animals much more easily than she can relate to the real world. One particular piece -the unicorn- is her favorite. Laura unconsciously identifies with the unicorn because, in their respective environments, they are both unique. Besides the parallelism between the menagerie's fragility and her own physical and emotional weakness, Laura is totally different from other girls, just as the unicorn is supposed to feel isolated because it does not belong in any known species.
When Jim, "the gentleman caller," accidentally drops the unicorn, its horn breaks. This turns it into an ordinary animal that will now be able to take its place in an imaginary animal community. Along the same lines, the fact that Jim has persuaded Laura to dance with him has turned her into an ordinary girl enjoying the simple pleasures of ordinary life.
While this is a fleeting moment, it somehow implies a turning point in Laura's life. She now knows that she can lead a normal life. It will be her decision to withdraw into her dream world once again or to take courage to fend for herself.
The play is called "The Glass Menagerie" because it is a unique thing about the main character Laura, which sets the main story of the play. Laura is no ordinary girl, she is handicapped, and does different things than most girls. She doesn't have gentleman callers, nor has a workforce skill. She often plays with her glass menagerie, a collection of glass animals, and plays records. Her differences are what cause her mother, Amanda, to think Laura will never wed or have a job, and will be miserable and lonely. The main plot of the story is to finding Laura a gentleman caller and meeting him. Which is why the name "The Glass Menagerie" is a fit title for the story.