First of all search for the desired Awesome icon here. In this example search for "stackexchange".

2. Click on the icon in the result list.

3. The HTML-code for the icon appears: ... class="fa fa-stack-exchange"....

4. Copy just the stack-exchangepart without the fa fa-. Concrete5 is taking care of it.

5. In your project, go to /application/config/generated_overrides/concrete.php.

6. You'll see a return array(... with some entries like 'site => 'sitename', 'version_installed => 5.7.5.1' and so on. After the last entries closing parenthisis and before the last closing parenthesis ); add the following code:

), // the last entries closing parenthisis

     // Don't forget the comma. Your code goes here:
     'social' => array(
        'additional_services' => array(
            array(
                'stackexchange', // the handle
                'StackExchange', // the title which will be shown in the dropdown
                'stack-exchange', // the icon name copied before
            ),
        ),
     )

); // The last closing parenthesis

That's it. Upload the file, empty site cache and you'll have a new entry in the Social Link Service Dropdown. Note that the Awesome icon needs to exist.