Custom options page checkbox will not save, despite working with text

Don’t comment out your validation code remember you are using it to validate the data, so its returning nothing right now, so its saving nothing, try this:

add_action('admin_bar_menu', 'make_parent_node', 999);

function make_parent_node($wp_admin_bar) {
    $args = array(
        'id' => 'test1234', // id of the existing child node (New > Post)
        'title' => 'test', // alter the title of existing node
        'parent' => 'new-content', // set parent to false to make it a top level (parent) node
        'href' => admin_url('admin.php?page=enter_timesheet')
    );
    $wp_admin_bar->add_node($args);
}

add_action('admin_menu', 'test_plugin_setup_menu');

function test_plugin_setup_menu() { /**/
    add_submenu_page('options-general.php', 'Forhandler options side', 'Forhandler Options', 'manage_options', 'mine-første-options', 'test_init');
}

/**/

function test_init() { /**/
    //echo "<h1>Hello World!</h1>";
    ?>
    <div>
        <h1>Codehero Dealers</h1>
        <form action="options.php" method="post">
            <?php settings_fields('mine_plugin_options'); ?>
            <?php do_settings_sections('mine-første-options'); ?>

            <input name="Submit" type="submit" value="<?php esc_attr_e('Save Changes'); ?>" />
        </form>
    </div>
    <?php
}

function mine_plugin_section_text() {
    echo '<p>Her finder du indstillinger til forhandler delen.</p>';
}

// add the admin settings and such
add_action('admin_init', 'mine_plugin_admin_init');

function mine_plugin_admin_init() {
    register_setting('mine_plugin_options', 'mine_plugin_options', 'mine_plugin_options_validate');

    add_settings_section('mine_plugin_main', 'Main Settings', 'mine_plugin_section_text', 'mine-første-options');

    add_settings_field('mine_plugin_checkbox', 'Forhandler checkout indstilling', 'mine_plugin_setting_string', 'mine-første-options', 'mine_plugin_main');

}

function mine_plugin_setting_string() {

    $options = get_option('mine_plugin_options');

    echo "<input id='mine_plugin_checkbox' name="mine_plugin_options[checkbox]" type="checkbox" value="1"" . checked( 1, $options['checkbox'], false ) . " />";
}

// validate our options
function mine_plugin_options_validate($input) {

    $newinput['checkbox'] = trim($input['checkbox']);
    return $newinput;
}

Rather than reading the option, setting up a conditional, and checking for the presence or absence of a value, we can use the WordPress checked function.