I don't know flapper, so I'm not positive this will work, but normally to reset an interval you need to assign the interval to a variable like this:
var myInterval = window.setInterval(function(){
var messages = ["INSPIRE", "EMPOWER", "EVOLVE", "ENCHANCE", "CHANGE LIVES"],
message = messages[Math.floor(Math.random() * messages.length)];
$('.flap').remove();
$(".ticker .fixedGuide h2").after("<input type='text' value='abcde' class='flap light S'>")
$(".flap").flapper(opts).val(message).change();
}, 5000);
$('.restartTicker').click(function() {
clearInterval(myInterval);
myInterval = window.setInterval(function(){
var messages = ["INSPIRE", "EMPOWER", "EVOLVE", "ENCHANCE", "CHANGE LIVES"],
message = messages[Math.floor(Math.random() * messages.length)];
$('.flap').remove();
$(".ticker .fixedGuide h2").after("<input type='text' value='abcde' class='flap light S'>")
$(".flap").flapper(opts).val(message).change();
}, 5000);
});