I will purchase a domain name for you if required.  You can also purchase your own name.  It better you buy one for yourself and control it, then if I did a runner, at least you can still have a website hosted for with the same domain name.  If you buy one yourself, but you will have to use the providers control panel to point to my servers.

If I buy a domain for you, it will initially be registered to me.  I will transfer the domain to you free of charge if required.  It will then be your responsibility to renew the domain.