You can choose any name that isn't already trademarked in your country. You don't need to register as a business in most countries. Legally in my country I am a sole proprietor under my real name trading as Evolutionary Casual Games. I know of another company that started using a similar name that's into software. So far, it hasn't mattered.
If your games become popular you can look at trade marking it. What does help is that search engines / social media allocate names on the first-come first-served basis.
Where I see a problem with your name is that people may forget there's a z at the end, and search for nice games. They're unlikely to find you because that term is too generic. You'd have to emphasise the Z somehow, like nicegamesZ.