There are several free games that are very interesting, but for some of these titles there doesn't seem to be any way to claim them by sending a donation to the developer. I'd really like to own some of these titles in the event that they for whatever reason get delisted. Is there a way to claim a free game that lacks a "Name your own price" feature?

I’d really like to own some of these titles in the event that they for whatever reason get delisted.

Unfortunately, if a dev decides to take their projects down, there is no way (regardless if you own it or not) to still have access to it. Developers are generally discouraged from doing that, as it makes a poor user experience, but Itch.io is an open platform, where the developers have more control over their creations.

If you are worried about losing access to a project, you can always download it and store it somewhere locally, which will guarantee you have access to it forever.

Is there a way to claim a free game that lacks a “Name your own price” feature?

The only way to claim a project, is to donate money to it. Developers can opt out of receiving money, for their own reasons, in that case I believe there is not really a way to claim such a project.

Hope that helps :)