It can be hard with bikes to speak with certainty about what name choice wins the title for most historically official or accepted, and it also always depends on how far back you want to count and where and in what langauges and what transalates to what and where and when etc, but "anchor bolt" probably gets it for all bolts that fasten gear and brake cables in terms of what the bigger name repair manuals and most manufacturers call it. There are others.
It is an M5 or M6 bolt, probably M6, and you'll need to evaluate yourself what length will work based on what you do to replace the plate, but probably like 10mm.
To replace the plate, use a sufficiently large steel washer, usually classified as a "large diameter" one. Consider adding another smaller one under the bolt, because a lot of bolts like this on brakes have a captive washer to start with, to do normal washery things. I like to try to find a thick washer and then use an appropriately sized needle file to put a little groove for the cable in it, which the stock plates all have. Do it in a way that doesn't create a stress riser, which is also why you're using a thick washer. This keeps it from wanting to rotate as you tighten it, increases the surface contact with the cable, and makes it hug more than flatten it. It doesn't have to be a big groove to accomplish this.
The most important thing when you do this is to safety test the heck out of it when you're done adjusting the brake but before it's ridden. Squeeze the lever ten times as though you were trying to destroy it. The main thing that can make replacing the plate with a washer not work is if the washer you choose is too hard of a material and allows the cable to slip. The above safety test is a good caution against that.
Some brakes need a more square-ish shape for the plate. You could use a file or dremel cut-off wheel to square down a washer in that case, or take it from a piece of flat steel strip stock.