First I think it is important to choose something that is possible to work with. I avoid really dirty ones. If the amount of soiling to an old rug is not acceptable to live with as is, I pass over that one, as I do not know how to do much cleaning of them. Barb Carroll tells me to lay a dirty hooked rug face down on dry snow out side for a day, then brush it off and much of the dirt will go with the snow. I have no snow so cannot say. I have used a rug cleaning product called Johnson’s glory, a foam spray which dries in a few minutes and can be vacuumed away taking some of the soil with it. But mostly I stay away from a dirty rug. Note, a rug hooked on monks cloth or modern rug linen can often be dry cleaned or washed by hand or gentle machine. The old ones on burlap will usually not survive this.
I am quite brave about attacking a rug with lots of tears or missing parts. Be sure your cost in any fragmentary ones is so low as to be negligible, then what can you loose by trying? Many truly antique rug fragments are worthy of considerable time and effort, and have a strong chance of being beautiful again. Judge your project on this matter and be sure the project is worthy.
One strong caution is this: Never use any adhesive of any kind on a hand hooked rug, not tape or glue or spray-on rubber backing.
Having decided you will mend it, determine the backing of the rug. Is it still pretty sturdy for the most part, or would the rug be better off faced completely with a new backing? If sewing or hooking through the old backing will cause it to break up, a whole new facing may be best.
If on the other hand the problems are just a few holes in the middle of the rug, those are easy to deal with. Just make a patch of new rug linen at least an inch and a half larger all around that the hole to be mended. Lay the new patch under the hole and using a large needle with strong thread, perhaps upholstery thread, sew the patch under the rug with a running stitch. I use a line of stitches about half an inch long close to the hole, and another line close to the edge of the patch. Think concentric circles here. When stitching, take care to stitch in between the loops of pile on the front of the rug, so the stitches do not show on the front side. You may want to widely whip around the edge of the hole on the top of the rug also before starting to re-hook. Hook material into the missing area, considering the height of the needed loops. Most old rugs were clipped on the surface, so match this as appropriate. This fill in material may be your familiar wool strips, or yarn (may be two strands at a time), or cotton strips cut from an old faded tee shirt. Many old rugs were made of cotton knit material. The tans are often cotton lisle stockings. Later rayon stockings and even silk ones were hooked into mats and rugs.
A different approach is needed when part of the edge of the rug is missing or much damaged. If a patch is needed for just a section of a straight edge, use a long narrow piece of new rug linen, and fold it longwise to make a patch larger than the missing section of border. Remove any old binding which is in your way. Sew the long folded patch under the missing or damaged area, with the folded edge out so it is a finished edge for the rug. Hook in the filling, then replace the binding or other edge finish.
Last we come to the rug which is just a fragment or is severely damaged or has a backing which is rotten and threatening to disintegrate! Form a new facing or backing for the rug using new rug linen. The size needed is about an inch and a half larger than the finished rug will be. With your fingers, lightly crease the linen where the finished edge of the rug will be. Do not get the backing too small or end up with the old rug bunched up and unable to lay flat. Using a long running stitch sew this hem in place.
Note: You might instead use a larger piece of new linen than will be needed, and turn in the hem only after you have secured the center and can closely judge the edge!
Place the old rug on it’s new backing with the raw edged hem hidden between them. Starting in the center, secure the two layers with pins or light basting here and there. Then making parallel lines of running stitches, secure the entire backing to the old rug, taking care to hide the sewing on the face of the rug, working down in the body of the hooking or in between the loops. Lay the work flat and check it often to keep it flat. last sew around the rug half an inch or so from the edge all around. On the face of the rug, whip down the edges of any holes or missing areas. When you have re-backed the rug, you can fill in with hooking as needed. You will find that the sewn in threads will get in the way somewhat of your re-hooking. This is why we use a long running stitch not a shorter prettier one. If that is a major problem, use a large needle with yarn to make high loops and replace some missing material that way. As long as the color is very close and the height of the new hooking is correct, the fiber is not as important as you might think.
Now the rug is ready for new binding. Use cotton twill rug binding tape if you like, or use wool strips. Plaid is pretty. Keep the binding muted and do not attract attention to it at the expense of the older part of the rug. The binding may be planned to show on the front also, by having 5/8 of an inch or more of unhooked edge to the backing.