Example of how to use jQuery to set the defautl value of the EE State dropdown. Change val(45) to the ID of the state you wish to use.

tw_ee_default_state_option.php

<?php
//* Please do NOT include the opening php tag, except of course if you're starting with a blank file

function tw_ee_default_state_option(){
    wp_add_inline_script( 
        'single_page_checkout',
        "jQuery(document).ready(function($){
            $('select.ee-reg-qstn-state').val(45).prop('selected', true);
        });"
    );
}
add_action( 'wp_enqueue_scripts', 'tw_ee_default_state_option', 50 );