The last name, "Glass," may carry with it any number of connotations, but one that comes readily to mind is fragility. Most of the Glass siblings are unable to live "normal" social lives: Franny is in a crisis and unable to adapt to college; Zooey hates most people he meets and judges them all cynically; Buddy is so antisocial he lives in a remote cabin and refuses to have even a telephone; and Waker is a priest far away in Ecuador.
When the Glass siblings try to maintain social lives or normal jobs, we know that it's a delicate balancing act. Franny and Lane's relationship is a key example of how difficult it is for members of this family to have outside relationship. Seymour's suicide is a constant reminder that all the Glass children are tenuously balanced on the edge of crisis. We happen to have joined the story just when Franny in particular has been tipped over the edge, but nearly all the Glass children's lives have an element of this fragility.
It's likely no coincidence that Franny and Zooey's names sound similar, given all that the brother and sister have in common: both face a choice about whether to pursue acting despite the spiritual poverty of the lifestyle; both were permanently changed by the education given them by their older brothers; both suffer from a tendency to judge everyone around them; both struggle with the self-torment driven by this very cynicism; and the two of them are "the only real beauties in the family." It's also significant that both names end in the diminutive "y," since Franny and Zooey are in fact the babies of their family.
Lastly, we can think about Mrs. Glass. Her children all call her "Bessie" instead of "mom." This actually tells us more about the Glass children than it does about Bessie herself. These precocious child geniuses seem as though they were never actually children at all. We understand this from the glimpses we get of the Glass children back when they were actually children. (Franny and Zooey were taught religious philosophy before they even hit puberty; Zooey had the vocabulary of a cult leader by the time he was twelve.) They don't call Bessie "mother" probably because they were never really children. This is, as always, just one interpretation – feel free to argue with us.
The peculiar Glass family is the focus of Franny and Zooey, and each member of the family is undoubtedly affected by the others. Franny and Zooey, of course, were fundamentally changed by their older brothers, Seymour and Buddy. Every member of the Glass family suffered through Seymour's suicide and was altered by it. It is Zooey (with some help from the spirit of Seymour and the "half-dead" ghost of Buddy) who helps Franny resolve her crisis.
Additionally, because the siblings share so many traits, the author can give us blanket info about the Glass family that serves to characterize more than one of its members. Buddy's claim, for example, that the Glasses featured in his story "speak a kind of esoteric, family language, a sort of semantic geometry in which the shortest distance between any two points is a fullish circle" is an apt description of both Buddy and Zooey, or direct characterization that offers a lot of bang for a little bit of text (Zooey.1.3).
It's fitting that Buddy is a writer, seeing as he's the one writing the story. (There are interesting questions as to how much authorial control Buddy has over his "characters," which we talk about in "Narrator Point of View.") This particular profession reflects Buddy's reclusive and somewhat meditative nature. Next up are Bessie and Les, who are retired Vaudeville performers. Remember that the effects of celebrity is one of the topics explored in Franny and Zooey. Bessie and Les's children went on to be child stars, and now one of them is a professional actor and the other is considering a similar career. Franny's struggle against egotism and her brother's struggle against narcissism is heightened by their decision to act (or not to act). Lastly, remember that Waker is a priest off in Ecuador, a reminder of how important religion is to the members of the Glass family. The fact that he is a Catholic priest proves Zooey's sentiment that "everybody in this family gets his goddam religion in a different package" (Zooey.6.87).