How can I braid my own shoelaces?
The name and way of making
The English word shoelace is a bit old fashioned, from the time strips of leather were used, which are still called lace in the knotting world.

Most shoelaces are braided these days.
The American shoestring is more appropriate, as long as you take braided string.

Modern shoelaces are braided, in the round, around a core for the round ones and without a core for the flat ones.

Braiding by hand
Here (my own site, not commercial) you find instructions how to braid with 8 strands, in the round. This is not the only site online with those instructions. If you want different ones, (or if the site is down) you can google for instructions.

As a beginner braider you will likely not be able to braid that many strands out of the hand, even as an experienced braider I have to work up to it.
If you want to work up to it, here you find several easier braids with which to work up to it.

Machines do not use just 8 strands, 16 strands are more likely, often 16 groups of two or more strands.

Using tools
If you want to braid 'proper' shoe laces by hand, you may want to look into working on a disk.
The disks are available under the name of Kumihimo disks.
'Proper' Kumihimo is done on a wooden frame in the shape of a stool.
This wikipedia page has a good photo of a traditional set-up as well as a good basic description of what the different items are.

And even with kumihimo disks or frames you will have to work up to a braid as intricate as shoelaces.


  

Braiding on a kumihimo frame by a young boy.
Photos cropped to protect anonimity. Photos by Willeke, free to use for all.

Making it fat and round
Whatever way you shape the braid, for the fat round laces you will also have to learn to braid around a core. This is a string of the right diameter or, more often used in commercial braids, a bunch of thin string that are kept together at the top of the braid and often suspended above or below for you to braid around.
In the photos there would be a set of strands suspended above the frame and allowed to drop into the braid at the same speed as braiding.
With a braiding disk you would hold the strands and move them around as you are braiding.
And with hand-held braids you would tie the core with the outer layer to your fixed point and braid around them. In the first link I give the core is mentioned in the instructions and is part of the pictures.

Weaving
Braiding with this many strands is closely related to weaving, the resulting string is a weave, but in the round with the strands on the diagonal.
If you would twist a series of strands around a tube, spiralling one way, you would be able to weave the other spiral in, by going over one, under one, all the way.
It will be hard to get it tight and neath and you would likely have to tighten the weave later.
The core would go through or instead of the tube.  

Having done the kind of work, I have to say that learning to braid is much easier.

Aglets, the hard bits at the ends
For 'real' shoelaces you also need to add aglets, I direct you to the site of Ian Fieggen to find out how to make them.

Would I do it
I would not make my own shoelaces, even though I have the knowledge and skills. Laces in the shops are relatively cheap, durable and available in many different variations and designs.
Making your own is going to be a lot of work, likely to result in lower quality and lower durability.
In fact, it is the other way around, I use commercially made shoelaces to train myself in braiding, they are very handy for that. Come in many nice colours and have aglets, the hard bits at the end that protect the ends and aid in feeding through strands to test the weaves.

How do I know
I am a member of the International Guild of Knot Tyers, whom do include braiding as part of our field of interest. My personal interest in braiding goes back at least 40 years, likely longer.
There is also an international organization for braiders, the Braid Society I am not a member but I do know several people who are.

PS, while I do not make my own shoe laces, I know people who make their own out of commercial string, see my answer on The Great Outdoors.