The phrasing will be familiar to those who participate in deliberative bodies which follow any of the many editions of Robert's Rules of Order, a system for parliamentary procedure, so-called because it is descended from the rules of conducting debate that originated in parliamentary government.
Robert's Rules are by leaps and bounds the most popular authority for parliamentary procedure in the United States, to the point where most people have heard of them, yet even experienced chairs may be unfamiliar with competitors like the Sturgis Standard (used by some academic bodies) or Demeter's Manual (adopted by some fraternal and labor organizations). A criticism of Robert's Rules is its use of old-fashioned or legalistic terminology, such as making a motion to previous question as opposed to close debate, or to order the yeas and nays instead of call for a voice vote. In my opinion, a call for the good of the order would qualify as well.
In any case, parliamentary manuals will have a set of rules for the order of business in a meeting, typically starting with a call to order, attendance, approval of minutes from the previous meeting, reports, and regular business. Then, the 11th edition says, chair may optionally call for
Good of the Order, General Good and Welfare, or Open Forum. This heading, included by some types of societies in their order of business, refers to the general welfare of the organization, and may vary in character. Under this heading (in contrast to the general parliamentary rule that allows discussion only with reference to a pending motion), members who obtain the floor commonly are permitted to offer informal observations regarding the work of the organization, the public reputation of the society or its membership, or the like. Certain types of announcements may tend to fall here. Although the Good of the Order often involves no business or motions, the practice of some organizations would place motions or resolutions relating to formal disciplinary procedures for offenses outside a meeting (Section 63) at this point.…
It is described thusly in Webster's New World Robert's Rules of Order Simplified and Applied:
Some organizations take time right before adjournment for the "good of the order." This segment allows members to give suggestions for improvement or to give compliments concerning the work of the organization. Usually business is not brought up during this portion of the meeting. Any ideas for new business that come from this segment are brought up at another meeting. However, if something urgent is brought to the attention of the members, a member can present it as a main motion during this segment. Until someone moves to adjourn the meeting, members can bring forward business.
In other words, this is the section of the meeting where people can bring up news or topics of general interest that are not on the main agenda for discussion. You don't need a full agenda item to say thanks for giving me a ride home after last month's meeting. I have seen it on some boards as a period for open discussion offering praise or criticism of officers or projects, and in others as the time to plug pet projects or ask for personal favors.
The phrasing is not found in the earliest editions of Robert's Rules, and may have been adopted from Freemasonry, where it appears in some rituals and oaths but is also found in casual use to refer to praise someone's contributions to the society. Good in the simple sense of something beneficial, and order thus refers originally to the Freemasons. This section of the agenda is often titled good and welfare, which also is found in Masonic texts.
But in the modern day, the phrase can also be read as order meaning the order of the day, in the narrow sense the daily agenda or the broader sense of the proper disposition of thing. You can also read it as order meaning organization, as that term can loosely be applied to any organized group or class without any connection to fraternal or religious groups. The more modern term might be open forum.
