What if I had a site named www.forexample.com and I no longer wished that name. I wanted whatever, www.thisexample.com

Can I do this and how?

Thank you very much!!

Go and register it (this being the domain name forum)

Or is the question about how you host and redirect the new name?

That's a simple question. Register the domain name you want as long as its available.

Register the new domain, put the DNSs pointed to your hosting account and transfer the content from the first domain to the second one

yes, you can do . First, you register your domain name which is new and redirect the old domain name ot the new domain. So, that every one will be knewing aobut the new domain name. But, you need to have really good hosting service provider

As usual you have some time to refuse domain name registration and you will lose about $1 and then using available funds register domain name you need to.
Good luck