InsaneJournal does have a mascot, and his name is Tweak.
Tweak was born a pixelated icon, which is now used in InsaneJournal's <lj user> tag. It was originally created by etteluap, but a request from the site's owner led to bri creating the logo now displayed on the top of InsaneJournal pages. A contest was held in January of 2005 to name this character, and Tweak won first place.
Tweak's appearance has changed over time as the default site scheme underwent changes. Originally, Tweak was placed in the 'Clean' scheme, which can be viewed here. When etteluap and surreal worked together to code a site scheme, Tweak's appearanced change. (The scheme was finished and, for the most part, coded by etteluap. The original design was a creation of surreal's.)
squeaky, the creator and owner of InsaneJournal, decided to give Tweak a personality. Tweak is supposed to represent all of the users of InsaneJournal, so the Tweak Says feature was invented. Here, any user is allowed to submit a message that will possibly be displayed on InsaneJournal pages. Submissions will be approved/rejected by administrators, and the text will be randomly generated each time you visit an InsaneJournal page (or by refreshing).
Thanks to tempest, Tweak is now available to be used as a moodtheme for your journal. You can find and use Tempest's Tweaks by visiting the Modify Journal page.