Select post from dropdown and add query args not working

Rename your <select> to something other than page_id.

The problem is that page_id is already taken (and handled) by WordPress (as query var).

This should do, for example:

<?php
if (isset($_GET['editevent']) && true == $_GET['editevent'])
    echo 'Post ID: '.$_GET['my_page_id'];
?>
<form method="GET" action="#">
    <select name="my_page_id" id="my_page_id" onchange="this.form.submit()">
        <?php
        $args = array(
            'posts_per_page' => -1,
            'post_type' => 'event',
            'season' => '2013-2014',
        );
        $posts = get_posts($args); 
        foreach ($posts as $post) :
            setup_postdata($post);
            ?>
            <option value="<?php the_ID(); ?>"><?php the_title(); ?></option>
            <?php
        endforeach;
        ?>
    </select>
    <input type="hidden" name="editevent" value="true" />
</form>