I thought for sure I had already tried this, but I guess I hadn't because it's working now. Makes sense -- you have to do the query before you can get the last id:
$sql = "INSERT INTO attendees SET firstName='$firstName', lastName='$lastName', address1='$address1', address2='$address2', city='$city', state='$state', zip='$zip', dayPhone='$dayPhone', evePhone='$evePhone', cellPhone='$cellPhone', fax='$fax', email='$email', typeOfAttendeeID='1', conventionID='1'";
$result = $db->query($sql);
$foreignAttendeeID = mysql_insert_id();
$sql2 = "INSERT INTO relationships SET role = 'mother', conventionID='1', attendeeID = '$foreignAttendeeID'";
$result2 = $db->query($sql2);