It's because the event is bubbling up to the DOM tree.
So you have to add next line at the beginning of your double click event handler :
e.stopPropagation();
Then it will look like this in your code :
function start_edit() {
$('.account_line').dblclick(function (e) {
e.stopPropagation();
var account_name = $(this).children('p').first().html()
$('#account_edit').css({
display: 'block'
});
$('#account_edit').children('#account_name_to_edit').html(account_name);
})
}
Description: Prevents the event from bubbling up the DOM tree, preventing any parent handlers from being notified of the event.