After I have designed a website, can I change it's name? How?

What do you mean change its name? The domain name (like: www.mydomain.com) or the name of the files?

Purchase a new domain and point the domain to the hosting server.

Tom is correct about purchasing a domain name, but if what you want is two addresses that point to the same website, then you want to do an auto redirect through your hosting company.

Isn't that what I said?

Point, redirect, it's all the same in DNS lingo.

Change from mywebsite.com to differentname.com

Then you just need to purchase a new one through your hosting provider.

I have designed a site on my desktop called mywebsite.com, I have not published it to the web. After reading some material on website names & searching I have decided it would be better to name it newname.com. I don't want to have to rebuild the site all over again with the new name. Is there a way I can rename it or clone it to a new name?

You don't have to rebuild it. It doesnt matter what the name of the website is on your desktop. You currently own: www.first.com, but you think it will be better to have www.second.com. All you have to do is buy www.second.com and put a redirect or point www.first.com to www.second.com after you purchase it. Then you don't have to reupload anything. But you would have to talk to your hosting provider on how to do the redirect. Some people have their own control panels, while others have to do it through their service provider.

Did you use a CoffeeCup product, such as HTML Editor or Visual Site Designer to create your site? I don't use VSD, so I can't really speak with authority about it, but, in general:

What you have on your desktop really doesn't matter, because it's a nonexistent domain name. When you purchase a domain name and upload the html and ancillary files to it, the url (essentially, the name) of the site becomes 'www.whatever-domain-name-you-bought.com"

So you should not have to worry about redirects. In fact, since the first domain doesn't even exist in the real world, nobody could get there to be redirected...

One thing to watch for: if the HTML in your site uses absolute references to what you say is the original 'website', they will have to be changed. If there are only relative references, you're ok.