If the hover is triggered by JS, just pause script execution via the keyboard. This is a much simpler way of freezing the DOM than the other answers suggest.

Here's how you do it in Chrome. I'm sure Firefox has an equivalent procedure:

  1. Open up Developer Tools and go to Sources.
  2. Note the shortcut to pause script execution—F8.

Pause script execution

  1. Interact with the UI to get the element to appear.

  2. Hit F8.

  3. Now you can move your mouse around, inspect the DOM, whatever. The element will stay there.