Installing the eyelets requires a staking tool, which clinches down the eyelet, an anvil, which supports the bottom side of the eyelet, and in a production environment you would use an eyelet press (staking tool on the top-right and anvil on the bottom-right in the image below). The eyelet is placed in the damaged through-hole and then crimped similar to the way a hollow rivet is installed in a ramp flap.
While the eyelets only cost about ten cents, the eyelet press runs about $1,300 and the staking tool and spring-loaded anvil will cost between $100-$300 each. Plus you need a staking tool and anvil for each size eyelet. Thankfully, there are a couple of cheaper options.
Link to Circuit Medic eyelet press and for our European friends a link to one manufactured by a Germain company, Bungard (much better price by the way).
Circuit Medic Eyelet Press
Bungard Eyelet Press
Best Electronics sells a plated through-hole repair kit, which costs about $200. The kit includes several sizes of setting tools and a basic anvil (non-spring-loaded), which are used with a hammer to clinch the eyelet.
Best Plated Through-Hole Repair Kit
Although you can put together your own kit for a lot less with the following products.
Mouser carries the Keystone setting tools. The Keystone 1714 staking tool is used with 1/16" to 3/32" diameter eyelets and the 1715 is used with 1/8" eyelets. Each costs about ten dollars.
Keystone 1714 Staking Tool
Keystone 1715 Staking Tool
You can buy a setting tool (staking tool on one side and anvil on the other) and tool base, which holds the anvil, at Circuit Medic. The setting tool (115-3120) costs $17 and the tool base (115-3122) costs $10.
This is not as good a setting tool as the Keystone since it does not roll-over the edge of the eyelet, so just use the anvil side of the tool. You would need two setting tools anyways since the staking tool and the anvil are used at the same time.
Circuit Medic Setting Tool
Circuit Medic Tool Base
Although I don't think the spring-loaded anvil is necessary, you can buy them at the Engineering Lab for $149. You could also use a cupped tip punch as the anvil, but the setting tool from Circuit Medic works better.
Engineering Lab Spring-Loaded Anvil
So for a grand total of $47 (plus some drill bits and a small hammer) you can do a professional eyelet repair.
Of course if you wanted to get fancy you could modify a bench press to hold the staking tool and anvil.