How do I convert an Appleworks database to something Lion will open?

You will have to open the database in AppleWorks to start. If it's possible to export it in .csv (comma separated variables) then you should be able to import it into s compstible database program that runs in Lion.

What database application is compatible with lion?

Hello,

You can use LibreOffice http://www.libreoffice.org/ (it's free but don't know if it will import the database)

or NeoOffice http://www.neooffice.org/ (free but $10 for an update ! ) , and read that page :

http://neowiki.neooffice.org/index.php/Opening_AppleWorks_documents

For more information about AppleWorks databases you can read these pages :

http://www.wilmut.webspace.virginmedia.com/notes/aw/page5.html

http://www.wilmut.webspace.virginmedia.com/notes/aw/page6.html