If a person attends five or six of these, he or she is worse than useless because the position is being taken up that might have been held by someone who would perform, and time is lost at meetings bringing that person up to date. Also, that amount of time invested does not allow a person to learn enough about an organization to contribute to policy decisions and action. On the other hand, attendance at nine, ten, or eleven meetings accelerates one's contributing ability. Then there is another time investment, which I recommend highly, and that is taking the Executive Director to lunch or having leisurely meetings in the office. These sessions can be helpful to a person's knowledge and performance because the conversation can be more informal and perhaps confidential.
Time is one of the important elements of effective board membership.
If you don't have the time, don't do it. I am honored, and I know what good work your agency does. However, I am on three boards right now, and up to my eyeballs in volunteer activities. I could not do the kind of job you want done, and although I hate to do this to a good friend, I just have to say no. The author would be most grateful for your responses and comments.