If you have a Pro account, you can add a Favicon to your site. This is a small icon that appears next to the site's URL in the browser's address bar, next to the site name in the Bookmarks list, and next to the page title on a browser tab.
Go to the Settings tab in the editor.
Click the Upload button that appears in the Favicon section and select an image to upload.
The image you select needs to be square and relatively small, as it will be scaled down to 16x16 pixels. If you get an upload error, try resizing the file to make it smaller. The icon should be in .jpg or .ico format.
When the upload is successful, you'll see both your Favicon and the option to remove it. Save and Publish to add the Favicon to the site.
If your Favicon does not show up on your site right away, try viewing your site in a different browser or on a different computer. Favicons are saved in a different part of the browser's memory than webpages so even after clearing your browser's cache, the Favicon may not show up for you at first.
New visitors will be able to see the Favicon the first time they visit the site and it should show up on your computer within a day or two at most. This is not related to Weebly, it's just how browsers work.