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.
Most VPS servers (including Digital Ocean) don’t include a host name. If you want a name, you’ll have to set one up separately.
You can request an 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.
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.