Scroll through the list and find the button you added. If you remove the check from the box next to the weather button's name, the button will remain installed but will no longer show up on the Google toolbar. You can it back to the toolbar at any time by accessing Toolbar Options and rechecking the box next to the button.
To get rid of the button so that it no longer appears on the list at all, click the Remove link.
If you just want to change some of the characteristics of the button, click the Edit link. For instance, you can edit the AccuWeather.com button to change the location for weather reports.
When done with your changes, click the Save button and the Toolbar Options window will close with the modifications saved. If you decide not to make any changes, you can click on the Cancel button to exit the window at any time.