The opening in the hex screw that holds the rear brake cable has become much looser than it used to be. What is the name of the part so I can order another one? (Hopefully they have some made of harder steel.)
Your picture shows a portion of a bicycle brake known as "direct pull" or "linear pull" or the Shimano label, "V-brakes." Utilizing a single cable, pulling the brake lever causes both arms to squeeze the braking surface of the rim.
The inner brake cable is held secure on one of the arms by the "cable fixing bolt" also termed "cable pinch bolt." The former is the term Shimano uses in their technical documents ( Shop Instructions and Dealer's Manual). This is generally an M6 x 12 mm hex cap bolt (sometimes labelled, hex cap screw, idk why as the threaded end is not pointed. Anyway, same thing). This is what we see in your photo.
You can spare yourself the damage to the hex head by using a 5 mm Allen key (the correct size for the M6 hex cap), making sure to fully insert the key into the hex slot. Keeping the key straight--in line with/parallel to-- the receiving hex hole and applying the torque perpendicular to this line helps keep the key fully engaged in the hex hole and puts the turning power on the largest area possible within the hex head. You may notice that a 3/16th Allen key seems to work good for these as well. Avoid using these because it is slightly small (4.75 mm) and when higher torque is needed to tighten, the smaller hex key can start to gall the inside of the hex cap. Eventually it becomes stripped as the interior angles are rounded. Using a badly worn hex key can also do the same thing.