All good ideas so far.
If you own the domain without any intent, I find that curious enough - though email alone is a fairly good idea in this era. Considering what happened to Yahoo mail and occasionally GMail, my own hosting service has a choice of 3 web based email apps which perform better.
Yet, there are LOTS of things to consider.
My wife has online friends - she's known some of them for 12 years, never met them in person. They met on forums about raising kids. As discussion became, shall we say more "involved" than the forums would tolerate, my wife decided to install the PhpBB application and host her own - JUST for her friends. This way they can take the discussion anywhere they like.
Same for online chat.
It's also her personal, self hosted massive photo site for family.
These are simply personal pursuits.
You can tap into your own interests and talents to form an exchange directly related to them. I know a luthier who uses his site both to educate people about the art and science of making violins, guitars and harps, but to use it as a medium of professional exchange from around the world.
If you're particularly tech savvy, there's a huge possible array of technical purposes it might serve. The basic feature is the public, static IP. When using private WiFi, public WiFi or other Internet connections, you often find yourself behind a firewall, unable to open a port the world can connect to, such that if you had reason to, say, serve FTP to someone who needs a file from you, it's quite a pain.
It's possible, of course, to use the hosting service own FTP server, but then your partner waiting for the file has to wait for your upload to finish before they can start the download - lest they beat you to the endpoint and never know if they get the entire file.
Instead, you can use the hosting service as a bridge for a live streaming connection between you and your partner for a more immediate exchange, even though you're being a firewall. This is something like what the various chat/messenger services do for you.
It's also your cloud. I mean YOUR cloud. That has value, if you use it intentionally, over leased cloud services from other sources. It's still a payment per month, and someone else's computer - but it's your server. You administer it.