I think you would be better off using a remodeling box (with the flanges that rotate out to grip the wall) and ignore the stud altogether. With a remodeling box, you just have to cut a hole big enough for the box, whereas if you use a regular box, you have to cut away more so you have room to nail it into the stud.
The main difficulties you'll face are that you don't know exactly where the lath is placed within the wall, and that the plaster will be brittle with age.
Start by using something sharp to gouge out a small hole in the center of the spot for your new receptacle; I use the pointed tip of a drywall saw. Once you've exposed the lath, expand the hole up and down until you find the edges of that piece of lath. The other lath are going to be approximately the same size and spaced up to maybe 1/4" apart. Adjust your planned spot up or down a little so that at least one edge of the box lines up with a gap in the lath -- it's easier to cut through the plaster on its own.
Expand the hole until it's big enough for the box, then use a jigsaw to cut away the lath. Vibration from the saw can cause more damage to the wall so take it very slow and easy. (Another reason to avoid the stud: if the lath ends at that stud, you'll only have about 1/2" of it to hold the plaster in place.)
Sometimes it's the simplest jobs that can be the most difficult. I'm willing to bet that its the summation of tiny errors leading to a visible flaw at the end. The question you want to ask is "How do I mark and drill a hole exactly where I want it to be?" (For a given value of exactly)
The answer to this question (and many others) is: Make a jig or paper template. Even if you're only doing it once, making a jig or a template is essential for precise work.
For this particular job, a paper template will probably do. Trace the outline of your hook (and holes) onto a sheet of paper. Now, at your leisure, using compass, straight-edge, GPS, and Lasers, find the center of the holes. Even if you just eyeball it, you're still more accurate than doing it on the wall, as you're working at a desk or table with good lighting and a comfortable working position.
Now, photocopy that sucker!
Cut it out, tape it to the wall, and start the drill hole with a small nail or even punch it with a nail set. You don't want the drill "walking" away from its start position. Go in with a small bit next. The idea is that the tool will do all the work so that you can concentrate on keeping that drill level, and on center. Go to your 1/4 inch bit for the plugs, and clean out the hole.
Remove paper, insert plugs, and you're golden.
It may seem like a lot of work, but the results are worth it.
1) Generally, you can't be accurate working vertically, so figure out a way to do the detail work on a flat surface.
2) Always start a drill hole with a nail or punch. It will always walk away when you power on the drill. (This is quadruply true on metal or hard plastics.)
Best Answer
If you can't repair the screw hole, the best option is to replace the box. If it's nailed into a stud and can't be easily removed, you might have to cut into it and destroy it. Once it is removed, you can replace it with an old work box mounted into the existing hole.