Host names are optional - you can always connect to a game using its IP address. But it’s nice to have a host name (like yourmush.somemuhost.com) so people can remember your game address more easily.
You can request a free AresMUSH sub-domain (yourmush.aresmush.com). See Subdomain Terms of Service for details.
You can also register a custom domain name (yourmush.com) for just a few dollars a year. There are many domain provider available. I like Namecheap personally, but any will work. Forbes has a list comparing some of the big ones.
Once you have a domain, you will need to create an “A Record” DNS entry and point it at the IP address of your droplet. The steps will vary depending on the domain provider. See Setting Up Domain Hosting With Namecheap for an example.