You'll need to get a unique domain to brand your links. With Rebrandly, you have a few of options:
Domain registrars - There are many options like GoDaddy, Domain.com, and HostGator.
Rebrandly - When you add a domain to Rebrandly, you have an option to search for one right from your account.
Add a subdomain to your existing domain - Using a subdomain makes your branded domain longer, but it is an option.
You can choose any of these options when you add your domain