What function or $user variable do I edit to manually increase a user's session timeout?
There are several modules that can do this for you.

https://www.drupal.org/project/session_cookie_lifetime
https://www.drupal.org/project/remember_me
https://www.drupal.org/project/persistent_login

I've had some problems with remember me and am still testing the others, but they give you more control over the session lenght.

By default drupals session expire after about 3 weeks, according to

ini_set('session.cookie_lifetime',  2000000);


You should be able to override this variable at database level as well, if you want to.