In case you've worked with repl.it before, you may have noticed that you are given a decently long subdomain for your project, like the following:
my-little-blog--dotcomboom.repl.co

You may have wanted to change that into a shorter, more memorable domain name, like this: mylittleblog.cf

Let's see how.

Getting your domain

This section will cover registering a domain name with Freenom. If you already have a domain, skip this part.

A word on Freenom

You don't own it

Freenom hands a selection of domain names out like free candy. If you've heard of dot.tk, it's these guys. In this tutorial I'll walk you through them, but I'd also like to point out that by registering a domain name with Freenom, you are still not entitled to it. It's perfectly valid and legal for them to all of a sudden deactivate your domain and use it to make money with ads, without notification. This is just a fair warning, some people have had domains from Freenom for a long time without any issues, but anything can happen. In short, nothing could be actually free. You are borrowing Freenom's domains for your purposes, and don't own them like you would from another registrar.

HTTPS

If you use Freenom, you won't be able to use HTTPS. You need to use "http://" before your domain instead. You will still be able to use your repl.co domain name if you need to use HTTPS, so make note of that. As https://repl.it/@turbio pointed out, HTTPS does work with a Freenom domain! You just need to wait a little bit of time for it to get set up. As for "mission critical" applications, like, say, Uptime Robot or something like that, you will probably want to use your repl.co domain.

Registering the domain

You will want to make or log into a Freenom account first. Then, you will be brought to the Client Area. Once you're signed in, you will find "Register a New Domain" under the Services menu.

This will bring you to Freenom's domain search. Type in the name you want. In this example, I searched for "mylittleblog". It will give you a list of what domains they have available.

Choose the one you want, then scroll down to the bottom and choose "Checkout". Don't touch the section in the middle, we'll cover that later. Choose the period you'll be using the domain, which is from 3 months to 12 months. (If memory serves, when you're going close to the due date, you are offered the opportunity to renew.)

Check the box to agree to the Terms and Conditions (reading them is a good idea too) and complete the order. Then, go back to your client area.

Attaching to repl.it

Repl.it setup

Go to your repl and click on the pencil icon next to the address in the preview pane.

Type in your domain name and if the domain is registered correctly, it will prompt you to add a CNAME record to your domain:

Copy where the record should point to, then go to your client area at your registrar (in my case, Freenom).

Registrar setup

From My Domains, go to the management page for the domain you chose. From Management Tools, choose Nameservers. Make sure it is checked to use default nameservers.

Then, find the area where you can add records; for Freenom, it is "Manage Freenom DNS". In the Name textbox, enter your domain name, nothing more, nothing less. Change the type to CNAME. Then, set the target to what repl.it asked you. Then, choose Save Changes.

Finish

Now go back to your repl. Repl.it now should have noticed that your domain is set up correctly and now lets you click on the Link button.

You're done! Your domain should now be attached to your repl.