Like all good names in the software industry, it started as an in-house code name.
We originally used Bugzilla for bug tracking and the developers in the office started calling it by the Japanese name for Godzilla, Gojira (the original black-and-white Japanese Godzilla films are also office favourites). As we developed our own bug tracker, and then it became an issue tracker, the name stuck, but the Go got dropped - hence JIRA!
Further investigation into the name has revealed that Gorira is Japanese for "gorilla", whilst Kujira is Japanese for "whale". So Gojira is roughly translated to mean "gorilla the size of a whale"! (Thanks to yusuke_arclamp — Oct 2002)
For those who care - it sounds best if you yell it loudly, as though charging into battle. C'mon - try it!