An update!
I had my first reminder to confirm my hostname or it would be deleted!
I think my phone line has been more stable recently, and the connection hasn't dropped out for a while, so I must have had the same IP address for 30 days (a new record).
The email notification from no-ip gave me 7 days to confirm the host name, or they will delete it. Clicking the link in the email went to a page where I was offered their basic paid service ($35 for 2 years or something), or a second button to continue with the free service. I elected to continue with the free service, and the next page gave me a recaptcha to confirm, and another button.
That's it.
All done for another 30 days (unless my connection drops during the meanwhile and I end up with a different IP address).