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:
- Open up Developer Tools and go to Sources.
- Note the shortcut to pause script execution—
Interact with the UI to get the element to appear.
- Now you can move your mouse around, inspect the DOM, whatever. The element will stay there.