Displaying a variable stored in functions.php inside widget

The widget operates in a different scope than the functions.php.

You could use two different approaches to get around that.

1. Make the variable global (put it into the top scope):

   // functions.php
   $GLOBALS['var_name'] = 'hello';
   
   // widget
   echo $GLOBALS['var_name'];
   

   But that is risky: any other script can change the variable now accidentally, and it is very hard to debug this.

2. Create a special class or function for the variable. You could even use one class or function to store many values. Example:

   class Theme_Data
   {
       private $data = array();
   
       public function __construct( $filter="get_theme_data_object" )
       {
           add_filter( $filter, array ( $this, 'get_instance' ) );
       }
   
       public function set( $name, $value )
       {
           $this->data[ $name ] = $value;
       }
   
       public function get( $name )
       {
           if ( isset ( $this->data[ $name ] ) )
               return $this->data[ $name ];
   
           return NULL;
       }
   
       public function get_instance()
       {
           return $this;
       }
   }
   

   In your functions.php, you can create an object now and add a value:

   $theme_data = new Theme_Data();
   $theme_data->set( 'default_posts_in_news_widget', 10 );
   

   In your widget, you can get that object and the stored value:

   // widget
   $theme_data = apply_filters( 'get_theme_data_object', NULL );
   
   if ( is_a( $theme_data, 'Theme_Data' ) )
       $num = $theme_data->get( 'default_posts_in_news_widget' );
   else
       $num = 5;
   

   You can even create multiple independent Theme_Data objects for different purposes, just create them with different $filter strings:

   $widget_data     = new Theme_Data( get_template() . '_widgets' );
   $customizer_data = new Theme_Data( get_template() . '_customizer' );