Would it have any other effect than clearing the search form of text, ie would it change any of the dropdown values to their default values as well?
Visually I don’t think there’s room for another button in the default UI, but you could certainly add one of your own, possibly by using the Widgets > Meetings Top area. A little jQuery should make it work, eg
$('button#my_reset_button').on('click', function(){
//reset form
$('#tsml #search input[name="query"]').val('');
$('#tsml #search').submit();
});
I haven’t tested it but something like that should work.