The first thing you'll want to do to add a weather widget to your site is create a new custom page, then select HTML, for more information on this first step check out this article: Making a Custom Page. Once you have your new page you should be looking at a totally blank slate like the image below (for this demo I named the page Weather):
From here you'll need to get the code to input a weather widget, the recommended way is to go through weatherwidget.io which offers a customizable and easy-to-use interface for creating your own widget.
After you've customized and created a widget you're happy with simply click the Get Code button and copy the code provided. From there paste it into the editor on your custom page save and preview.
Enjoy!