This will require a little bit of visualisation on your part. TODAY I found that the name is "Door Check Arm."

Where the hinges are on the front driver's side door, there is a small arm which helps hold the door open. Mine appears to be loose or malfunctioning.

Anyone know about this part, where a replacement may be obtained or how to fix it?

 

Jack, I have seen three of the X300 cars that, on the drivers' door, had this issue. In all three of the cases it turned out that the check arm was mounted to the door sheet metal and that the door had actually fractured where the arm attached. I don't know that this is the case with your car, but if it is it is not a big deal to fix. Pull the door panel and look in there to confirm. If that is your problem you can weld the steel back together without too much drama. Provided, of course, you have both the skills and the welding equipment. Mig would be my choice. Good luck with this.