This article suggests a few types of insights and reports you can generate by applying analytical methods and techniques to your business intelligence platform (assuming it's a centralized one that stores all this data): http://www.sisense.com/blog/bi-on-bi-analyzing-your-analytics

Do you have any suggestion for analyses you can perform on the usage, adoption or other elements of your business intelligence system?

(Disclosure: I am the editor of the blog in which this article was posted)