checkbox in box_product

I put a checkbox called personalization in the box_product:

The only thing worth noting regarding posting is if you use http post protection then a ticket is provided with form_draw_form_begin() that is required to submit a post. But I don't think that is your problem here.

I think that you can try to do this at this point:

$('body').on('submit', 'form[name=buy_now_form]', function(e) {
(...)

jQuery on() is necessary for placing events in any ajax retrieved content that is not available at first page load.

Yes, but you don't need this on page load (or i don't understand question :P). You can pass your personalisation data here:

(...)
data: {
    'data': $(form).serialize() + '&add_cart_product=true',
    'personalisation' : ...
    },
(...)

...and in pages/ajax/cart.json.inc.php save it to session or something ;)

Second thought:

Catalog -> Option groups -> Create new group (Name: Personalisation, Type: textarea)

Hide "Peronalisation" container with something like this:

$('textarea[name=options[Personalization]]').parent('.options').css('display', 'none');

Now:

Personalise 

Further:

$("input[name=blabla]").click(function() {
    if (this.checked == true) {
        $($('textarea[name=options[Personalization]]').parent('.options')).slideDown();
        $('textarea[name=options[Personalization]]').prop('required', 'required');
    } else {
        $($('textarea[name=options[Personalization]]').parent('.options')).slideUp();
        $('textarea[name=options[Personalization]]').removeProp('required');
    }
});