what is that little metal bar that keeps a car door open?

The part that holds the car door open is the detent rod, aka check rod. Some people refer to it as the door hold-open mechanism. It has 2 detent positions which hold the door half open or fully open.

EDIT: The repair expense will vary depending on the make and model of the vehicle but it is typically very inexpensive. Depending the the type of detent mechanism, the parts kit can cost anywhere from $15 or $20 to $40 or $45 and the labor charge will usually be 1 hour at whatever the hourly labor rate is at the shop. It doesn't take an hour to do the repair on most vehicles but shops usually have a 1 hour minimum labor charge.

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