Add a checkbox to post screen that adds a class to the title

Yes, that’s right, you need a Custom Meta Box that’ll create the checkboxes in the Post Editing screen.

In this case, they are radio buttons, as we need only one value.

The following is the code that creates it. Put it in your theme’s functions.php file or create a simple plugin so this becomes theme independent.

/* Define the custom box */
add_action( 'add_meta_boxes', 'wpse_61041_add_custom_box' );

/* Do something with the data entered */
add_action( 'save_post', 'wpse_61041_save_postdata' );

/* Adds a box to the main column on the Post and Page edit screens */
function wpse_61041_add_custom_box() {
    add_meta_box( 
        'wpse_61041_sectionid',
        'Title Color',
        'wpse_61041_inner_custom_box',
        'post',
        'side',
        'high'
    );
}

/* Prints the box content */
function wpse_61041_inner_custom_box($post)
{
    // Use nonce for verification
    wp_nonce_field( 'wpse_61041_wpse_61041_field_nonce', 'wpse_61041_noncename' );

    // Get saved value, if none exists, "default" is selected
    $saved = get_post_meta( $post->ID, 'title_color', true);
    if( !$saved )
        $saved = 'default';

    $fields = array(
        'red'       => __('Red', 'wpse'),
        'green'     => __('Green', 'wpse'),
        'blue'      => __('Blue', 'wpse'),
        'default'   => __('Default', 'wpse'),
    );

    foreach($fields as $key => $label)
    {
        printf(
            '<input type="radio" name="title_color" value="%1$s" id="title_color[%1$s]" %3$s />'.
            '<label for="title_color[%1$s]"> %2$s ' .
            '</label><br>',
            esc_attr($key),
            esc_html($label),
            checked($saved, $key, false)
        );
    }
}

/* When the post is saved, saves our custom data */
function wpse_61041_save_postdata( $post_id ) 
{
      // verify if this is an auto save routine. 
      // If it is our form has not been submitted, so we dont want to do anything
      if ( defined( 'DOING_AUTOSAVE' ) && DOING_AUTOSAVE ) 
          return;

      // verify this came from the our screen and with proper authorization,
      // because save_post can be triggered at other times
      if ( !wp_verify_nonce( $_POST['wpse_61041_noncename'], 'wpse_61041_wpse_61041_field_nonce' ) )
          return;

      if ( isset($_POST['title_color']) && $_POST['title_color'] != "" ){
            update_post_meta( $post_id, 'title_color', $_POST['title_color'] );
      } 
}

^{code based on this Answer}

How to use it in the theme

With the proper CSS rules in place (h1.default, h1.red, etc), wherever you want to apply the color class to the title (index.php, single.php, etc), use something like:

<?php $title_color = get_post_meta( get_the_ID(), 'title_color', true); ?>
<h1 class="entry-title <?php echo esc_attr($title_color); ?>"><a href="<?php the_permalink(); ?>"><?php the_title(); ?></a></h1>