You could use RAISE_APPLICATION_ERROR like this:
DECLARE
ex_custom EXCEPTION;
BEGIN
RAISE ex_custom;
EXCEPTION
WHEN ex_custom THEN
RAISE_APPLICATION_ERROR(-20001,'My exception was raised');
END;
/
That will raise an exception that looks like:
ORA-20001: My exception was raised
The error number can be anything between -20001 and -20999.