The advantages are similar to those for using your own domain in an email address:
Be sure that you've added a CNAME record in the DNS to point your domain to www.own-id.com. Domain forwarding won't work.
If you've set up your own domain name with Own-Id, you can change the OpenId identifier it delegates to by following these steps:
Own-Id follows the OpenId 2.0 specification to allow you to set up your own domain name as an OpenId Identifier. Say you own the domain name example.com, and you want to use id.example.com as your OpenId identifier. You create a CNAME DNS record pointing id.example.com to www.own-id.com, then you visit http://id.example.com/. This comes up with a page asking you to enter an OpenId which you can get from a provider. Own-Id stores in a database the association between your domain name and the OpenId identifier. Then when you log in to a website using id.example.com, the website does a lookup behind the scenes according to the OpenId specification and uses the Identifier you got from the provider.
Since Own-Id only stores information which is public anyway (your domain name, OpenId that it's delegated to, OpenId server), there isn't really a privacy issue. If you can think of a privacy concern, please let us know.
Let's say that Own-Id suddenly disappeared, never to return. As a user, this would mean that you'd be unable to use your custom id (eg. id.example.com). What could you do? Here are your options in order of increasing hassle:
I should say that we've no intention of going out of business, and Own-Id is a highly reliable service.
If Own-Id turned evil, what could they do? I can't think of anything we could do that would be any worse than the service simply not working. If you can think of anything, let me know. Our project is released as free software, so feel free to inspect the source code.
I think the easiest method of delegating your OpenId depends on several factors:
If the answers 1-4 are 'yes' and the answer to number 5 is 'not very', then it's easiest to go ahead and edit the html. If your DNS hosting company has an easy web interface (like GoDaddy) and you're a bit shaky on the other questions, then I'd suggest that www.own-id.com is for you.
Please send me any comments and questions.