Step 1: Starting Off
First you should get hold of some extremely fine plastic line. Fishing line would probably work, as long as you can find one thin enough. I'm using a plastic line that's meant for sewing beads and such to clothes, it's about as fine as the hair on my scalp. This would probably also work with regular fabric thread, the key is to have it as fine as you possibly can get hold of. Even though thicker thread is stronger we're going to use it in a way that makes the thinner thread both strong and durable.
Second thing you're going to need superglue, the faster it dries the better. It's also easier if you can find a superglue with a brush already in the bottle as it will facilitate steps further down the road.
Third, and final, an X-Acto knife for all your cutting needs. Though, a scalpel or a really sharp knife, or even scissors, would work too.
Step 2: Wrapping the Shoelace
Take a length of the plastic line, as long or short as you deem necessary (at least 12 inches I'd say though). Since we're going to wrap this line over the shoelace to both crimp it and give it rigidity it helps if you can attach it first and just concentrate on wrapping. As you can see from the top illustration on the image below you could make a simple slipknot like this, thread it over the shoelace and tighten it. From there it's easy just to wrap, and gives a rather neat result as well.
The bottom illustration shows what we're after. A nice uniform wrap, though not necessarily as spaced out like that, all across the length of what will be your new aglet. Doing one lap is a bare minimum, but if you have enough plastic line and enough patience going back over it a second time is definitely recommentable. The more neat you are the better the end result will be. Though, if you're like me and care more about it working than looking extremely good then it will work just as well even if you wrap it chaotically. The aim is to get good coverage and good crimping.
As you can see I haven't wrapped all the way out to the end both because it's hard to wrap a shoelace that far out nearing the end and because of the damage the shoelace has received the new aglet will look better with the damaged parts removed. Basically, if you can sacrifice less than an inch overall on your shoelace then do it, the end result will be better.
Attach the plastic line after wrapping with a knot and move onto the next step.
Step 3: Glue
What you need to do after wrapping is to glue the plastic line in place. Here's why it helps to have a brush in your superglue bottle. Basically cover all the plastic line with glue uniformly, making sure that you missed no part and making sure you don't cover more than you have to. Though, a little spillover towards the frayed edges isn't exactly a problem.
Set this aside to dry, or gently roll it between your thumb and index finger to stimulate drying. A word of caution though, this will cause your fingers to be encrusted.
Step 4: Almost Done
Once dried, you're done. There you have an aglet that will last you a lifetime, though no guarantees for the rest of the shoelace.
There are methods here on Instructables.com that recommend using shrink tubing to make new aglets. It looks absolutely smashing, and works really well, but they're not durable at all as I've found out. These superglued aglets are...and there's still nothing stopping you from using shrink tubing over this to brighten up the color.
Step 5: Last (optional) Step
The method I described can be utilized basically on any type of shoelace or cord, as long as it's suitably thin. If you use para-cord and don't want to burn the edges this is a good alternative.
If you have wider shoelaces, like the black one pictured below, it helps to fold it in half before you start wrapping with the plastic line. It gives a neater result. Also, if you use black shoelaces there's nothing stopping you from breaking out that Sharpie and color the new aglets to match the rest of the shoelace.
That's it for me, I hope someone had use for this.