There are two ways to accomplish this.
- Click on the Dev Tools popup so the window is focused and then press F5. This will reload the popup and Dev Tools, and will not cause the Dev Tools window to close.
- If that doesn’t work for you, go to
chrome-extension://extensionid/path/to/popup.html
on a separate tab on Chrome. From there, you can inspect element, and refreshing the file will not cause Dev Tools to close.