Host names are optional - you can always connect to a game using its IP address. But it’s nice to have a host name so people can remember your game address more easily.
Some MUSH hosts may give you a host name (like yourmush.somemuhost.com) but Digital Ocean doesn’t. If you want a name, you’ll have to set it up.
You can also request an AresMUSH sub-domain (yourmush.aresmush.com) by contacting me directly. AresMUSH sub-domains are given out at my sole discretion, and at no charge.
You can also register a custom domain name (yourmush.com) for just a few dollars a year through sites like NameCheap. Once you have the domain registered, use the Digital Ocean Host Name Tutorial to set up your domain.