Make few changes to this code and add it to your site collection master page and it should work. I didn't tested it yet, so you better go through code before implementing it.
var siteUrl = '/sites/MySiteCollection';
var serverRelativeUrl = '/sites/MySiteCollection/Default.aspx';
function updateWebPartTitle() {
this.clientContext = new SP.ClientContext(siteUrl);
var oFile = clientContext.get_web().getFileByServerRelativeUrl(serverRelativeUrl);
var limitedWebPartManager = oFile.getLimitedWebPartManager(SP.WebParts.PersonalizationScope.shared);
this.collWebPart = limitedWebPartManager.get_webParts();
clientContext.load(collWebPart);
clientContext.executeQueryAsync(Function.createDelegate(this, this.changeTitle), Function.createDelegate(this, this.onQueryFailed));
}
function changeTitle() {
if (!collWebPart.get_count()) {
alert('No Web Parts on this page.');
}
var oWebPartDefinition = collWebPart.get_item(2);
this.oWebPart = oWebPartDefinition.get_webPart();
var today = new Date();
var dd = today.getDate();
var mm = today.getMonth()+1; //January is 0!
var yyyy = today.getFullYear();
if(dd<10){dd='0'+dd} if(mm<10){mm='0'+mm} today = mm+'/'+dd+'/'+yyyy;
oWebPart.set_title(today);
oWebPartDefinition.saveWebPartChanges();
clientContext.load(oWebPart, 'TitleUrl');
clientContext.executeQueryAsync(Function.createDelegate(this, this.onQuerySucceeded), Function.createDelegate(this, this.onQueryFailed));
}
function onQuerySucceeded() {
alert('Title changed for Web Part: ' + this.oWebPart.get_titleUrl());
}
function onQueryFailed(sender, args) {
alert('Request failed. ' + args.get_message() + '\n' + args.get_stackTrace());
}
Note:
Here's the references I picked this code from,
How to: Work with Web Parts on a Page Using JavaScript
Get Current DateTime using JS