How do I pass arguments to dashboard widget callback functions?

The args are stored in the 2nd variable passed to your callback function.

add_action( 'wp_dashboard_setup', 'sample_widget_setup' );

function sample_widget_setup() {

    wp_add_dashboard_widget(
      'sample_dashboard_widget',
      'Sample Widget',
      'sample_dashboard_widget_callback',
       null,
        'sample_string'
    );
}

function sample_dashboard_widget_callback( $var, $args ) {
    var_dump( $args );
}

Output from above:

array
  'id' => string 'sample_dashboard_widget' (length=23)
  'title' => string 'Sample Widget' (length=13)
  'callback' => string 'sample_dashboard_widget_callback' (length=32)
  'args' => string 'sample_string' (length=13)