Max Anderson, the Director of the Indianapolis Museum of Art has thought a lot about museum metrics. His museum's "Dashboard" is a model of clarity and depth of information:
His article The Metrics of Success in Art Museums is the best summary of different kinds of museum metrics. It is available through the Getty Leadership Institute web site, where a number of other good articles are also available.
Here is an excellent, if academic, overview of thinking about the issue of measuring success in museums, which includes Anderson's work: Achieving excellence: Investigation into the use of performance indicator in museums by Alessia Zorloni.
The Visitor Studies Association has long been a leader in helping museum understand visitors, although their focus is more often on exhibit evaluation than overall measures of success:
The museum associations have also been active in this area, establishing benchmarks to use. One example is the Association of Children's Museums' Children's Museums Metrics Reports, which proived a massive amount of data. Jim Collins in Good to Great and the Social Sectors puts his finger on the core issue:
The critical question is not "How much money do we make?" but "How can we develop a sustainable resource engine to deliver superior performance relative to our mission?"
The field of museum metrics has a long ways to go, but it is also much more sophisticated than one might suspect. The biggest issue is that every museum needs to find benchmark museums that are truly comparable (often a challenging task) and establish metrics for its own success that are related to its unique mission and vision.