To change the icon displayed for a widget in a Widget Controller, simply click the icon in the Settings panel. You can upload your own icon with the plus sign at the bottom of this window. The box (black by default) in the lower right lets you pick the icon's color.

Rename titles by clicking the title of the widget in the settings panel.

Similar options are available for renaming or changing icons for sections and pages. (Just click on names and icons. You'll find them.)