How to pick the default selected value in wordpress dropdown?

@Antti Koskinen Sorry I missexplained, this is not coming from Elementor, it is just that I do not know where to put the code :)

I have modified the file class-wp-widget-search.php but it did not have any effect. I just added this to the file :

'selected' => 2,

Here is complete file code :

/**
 * Sets up a new Search widget instance.
 *
 * @since 2.8.0
 */
public function __construct() {
    $widget_ops = array(
        'classname'                   => 'widget_search',
        'description'                 => __( 'A search form for your site.' ),
        'customize_selective_refresh' => true,
        'selected' => 2,
    );
    parent::__construct( 'search', _x( 'Search', 'Search widget' ), $widget_ops );
}

/**
 * Outputs the content for the current Search widget instance.
 *
 * @since 2.8.0
 *
 * @param array $args     Display arguments including 'before_title', 'after_title',
 *                        'before_widget', and 'after_widget'.
 * @param array $instance Settings for the current Search widget instance.
 */
public function widget( $args, $instance ) {
    $title = ! empty( $instance['title'] ) ? $instance['title'] : '';

    /** This filter is documented in wp-includes/widgets/class-wp-widget-pages.php */
    $title = apply_filters( 'widget_title', $title, $instance, $this->id_base );

    echo $args['before_widget'];
    if ( $title ) {
        echo $args['before_title'] . $title . $args['after_title'];
    }

    // Use current theme search form if it exists
    get_search_form();

    echo $args['after_widget'];
}
/**
 * Outputs the settings form for the Search widget.
 *
 * @since 2.8.0
 *
 * @param array $instance Current settings.
 */
public function form( $instance ) {
    $instance = wp_parse_args( (array) $instance, array( 'title' => '' ) );
    $title    = $instance['title'];
    ?>
    <p><label for="<?php echo $this->get_field_id( 'title' ); ?>"><?php _e( 'Title:' ); ?> <input class="widefat" id="<?php echo $this->get_field_id( 'title' ); ?>" name="<?php echo $this->get_field_name( 'title' ); ?>" type="text" value="<?php echo esc_attr( $title ); ?>" /></label></p>
    <?php
}

/**
 * Handles updating settings for the current Search widget instance.
 *
 * @since 2.8.0
 *
 * @param array $new_instance New settings for this instance as input by the user via
 *                            WP_Widget::form().
 * @param array $old_instance Old settings for this instance.
 * @return array Updated settings.
 */
public function update( $new_instance, $old_instance ) {
    $instance          = $old_instance;
    $new_instance      = wp_parse_args( (array) $new_instance, array( 'title' => '' ) );
    $instance['title'] = sanitize_text_field( $new_instance['title'] );
    return $instance;
}

Thank you.

To set the default value for the category dropdown, you need to pass it as a parameter for the function. Like so,

$dropdown_args = array(
  'selected' => 2,
);
wp_dropdown_categories( $dropdown_args );