<input type="hidden" name="firma" value="" id="firma" />
$(document).ready(function() {
    var $select = $('#select'),
    $custom = $('#custom'),
    $customValue = 'Другое',
    
    $select.on('change', function() {
       if  ($('#select :selected').val() == $customValue) {
         $('#custom').show();
        }
else {
$('#custom').hide();
}
    });

 $('#form').on('submit', function() {
         if ( ($('#select :selected').val() == $customValue) && ($custom.val() != '') ) {
           $('#firma).val() = $custom.val();
        }
      else{
      $('#firma).val() = $('#select :selected').val();
      } 
       
    });
});