I use the following syntax to get a list of tables in the SQLite database to which FDConnection1 is connected:

FDConnection1.GetTableNames('', '', '', ListBox1.Items, [osMy, osSystem, osOther], [tkTable, tkView]);

But the only table that gets populated in ListBox1 is sqlite_master.  

Any thoughts would be appreciated.  Thanks.

If you expect, that there are other tables, then try to query any one of these tables.
If you will fail to query, then probably this is a new database. Which may be created
when you specified wrong path to existing database.

If all above is not true, then please provide environment report, generated from your
test application:
http://docwiki.embarcadero.com/RADStudio/XE7/en/DBMS_Environment_Reports_(FireDAC)#Using_Delphi_Code


-- 
With best regards,
Dmitry Arefiev / FireDAC Architect