From here, you should be able to manually change both your first and last name.
After digging deeper, it seems that the issue is related to the way the Skidrow version of the game is patched or cracked. The Driver Name Changer tool relies on certain files or code within the game to function properly, but the Skidrow version seems to have some differences that prevent the tool from working correctly. F1 2013 Driver Name Changer Fix-Skidrow