Just ran into an issue, and support said to throw it in as a feature request.
When monitoring a server with AppInsight for IIS, if you rename a website in IIS, AppInsight doesn't pick up the change. It still shows the old site name and says the site status is "Unknown".
The only current way to resolve it (aside from likely some SQL tampering) is to remove and re-add the AppInsight template to the server.