What you're paying for
You're renting the name, a year at a time, from a registrar. As long as you keep paying, nobody else can have it. Stop paying and it goes back on the market, usually to somebody who resells it.
The yearly fee is small enough that it's never the interesting number in a website decision. What matters is that the renewal card doesn't expire and the reminder email goes somewhere you read.
Why the name on the account matters
Everything hangs off the address. Your email, your website, the link on your Google listing, the address on your truck and every business card you've handed out. Whoever controls the account controls all of it.
The story that goes wrong is always the same one. A web guy registered it for you years ago as a favor, and now he doesn't answer, or he answers with a number. You're not arguing about a website at that point. You're arguing about your phone number's online equivalent.
So register it yourself, in the business name, with your own card and your own email. It takes ten minutes and it never has to be dealt with again.
Picking one
Short, sayable over the phone, and your business name if it's available. A name a customer can hear once and type correctly beats a clever one every time.
Skip the address stuffed with a job and a town. It stopped being a shortcut with search engines a long time ago, and to a homeowner it reads as an ad rather than as a company. Your own name is the one that grows with you, and it's the one that still fits when you add a second trade or a second town five years from now.
Who does the work of pointing it
The address and the website are separate things. You own the address; somebody has to tell it which computer to send visitors to. That's a settings change, it takes minutes, and it doesn't mean handing anybody the account.
Ask for it that way: keep the account, share the access needed to point it, and take the access back when the work is done. Anyone who insists on owning the address to do the job is telling you something about how the relationship ends.