The contact "Display name" field exists in Thunderbird but not on the iPhone. When you enter a new contact in Thunderbird (First name, Last name), the "Display name" is automatically filled with the value "First name Last name".
If you don't want to have an empty "Display name" field in your contacts when synchronized from the iPhone, it's possible to automatically generate this "Display name" with BirdieSync. Open BirdieSync options in Thunderbird (menu "Tools/BirdieSync/Options
..."), "Contact" panel, "Miscellaneous" tab and for the setting "'Display name' synchronized to Thunderbird" choose "First name Last name" or "Last name, First name". Then contacts synchronized from the iPhone will have a "Display name" generated according the option you chose.
Note that the "Display name" is not mandatory to order and display the contact list in Thunderbird. If you don't need to assign specific values in the "Display name" field, different from the First name and Last name, you may simply ignore the "Display name" field and change the way the contact list is displayed. To do this, open Thunderbird address book, then select menu "View/Show Name As" and select "First name Last name" or "Last name, First name".