Some people forget to do so and when the host goes belly up or the client (you) want to move to a better service you find out you do not own the domain name. The name you spread around the internet promoting your product.
The price of a name is cheap like $10 a month or less. Always use a search engine to search and not a web host as they have been known to buy the domain seconds after you pass on it that moment. You find it for sale at a higher price a few minutes later.
Also have copies of all the website graphics and pages.
Never rely on the website creator or host to turn over the files.
Learn how to upload your website pages yourself. Also setting up the email accounts. Even if you have someone else do the work. It is not that hard. It becomes hard when something goes wrong and you cannot contact anyone.
You could fix most issues.
I use GoDaddy http://www.http://godaddy.com and to host my site I use LunarPages http://www.lunarpages.com
GoDaddy does try upselling you on all their pages. I recently had to pay my yearly fee for my domain names and it was irritating the number of attempts. Never had issues with LunarPages.
Again do not do a search for a domain name at these or similar sites or you will lose that name.