JavaDB aka Derby aka Cloudscape would be a decent choice; it's a pure Java SQL database, and it's included in the JRE, so you don't have to ship it with your code or require users to install it separately.
(It's actually not included in the JRE provided by some Linux package managers, but there it will be a separate package that is trivial to install)
However, Derby has fairly poor performance. An alternative would be H2 - again, a pure Java SQL database that stores a database in a single file, with a ~1MB jar under a redistributable license, but one that is considerably faster and lighter than Derby.
I've happily used H2 for a number of small projects. JBoss liked it enough that they bundled it in AS7. It's trivial to set up, and definitely worth a try.