<p>I'm using Forms Authentication (and the built in Login controls) and letting users click the "remember me" checkbox when logging in. When they return to the site, they are automatically logged in as expected. Is there an event I can catch when this happens? I've been studying the FormsAuthenticationModule.Authenticate event, but I'm not sure that is the correct approach.</p>

First of all, look into using the Login control that's available in 2.0+ (as I remember).  There is a Authenticate event you can handle on that control, which is where you can set your session variable.
Additionally, let's assume after authenticating the user is taken to some "home" page.  So you can set your session variable inside the "home" page's Load event handler.