To have your own website you must have a unique domain name. Thousand of domain name is being reserved everyday. So you may not get your desired domain name free that usually occurs. In case of such situation you have to choose the alternate domain name nearly relevant to you fist choice. you may use "-" between two words. Or there are many extension such as .net,.info etc available you may choose the best one that suits you best. There are many site online that provide that provide free domain name if you purchase hosting from them and offers many special offer. You may also visit their services and then take final decision.
If you get a domain name, it is best to register it yourself and in your name. There are reports that some web host who provides free domains would register under the web host name and later blackmail the customers if they want to leave. Domain names are cheap in today's context so there is no reason not to do it yourself and save you all the trouble in the future.