How’s a “duration picker” called and does it exist?

You could use a set of customized jQuery spinners

 $('#seconds').spinner({
     spin: function (event, ui) {
         if (ui.value >= 60) {
             $(this).spinner('value', ui.value - 60);
             $('#minutes').spinner('stepUp');
             return false;
         } else if (ui.value < 0) {
             $(this).spinner('value', ui.value + 60);
             $('#minutes').spinner('stepDown');
             return false;
         }
     }
 });

like this:
http://jsfiddle.net/xHzMw/1/