I have a ‘vanity’ domain name and a web site to go with it. It’s hosted by a big-time hosting place. It’s a static HTML/PHP page, but I’ve also done ‘vanity’ sites by using a simple WordPress theme.
Although you can host on your own server at your house, there are security risks involved, along with maintenance. First, do you have a dedicated IP address with your ISP (which will cost extra, although there is some geeky stuff you can add to do that)? If not, you’ll need to do additional tricks to ‘point’ your domain to your server. And there are some internal firewall/port forwarding you will have to do locally, and protection for other devices/users on your internal network. A lot of security risk if you don’t do it right.
This is all ‘fun’ for geeky-types, and might be worth the effort. But I think the risks and work is a big consideration for self-hosting. You can purchase a domain name (which you will need anyhow) and cheap hosting for a couple $ a month. And that lets the hosting place worry about all the security and updates. You just worry about the content.
Some hosting plans will allow you to have multiple domain names on your hosting plan (I’ve got 20-40 domains on mine, but I’ve upgraded the hosting plan).
Even though I’m a geek, I’d rather the hosting place both with that stuff. I’ll geek on the content.