New or Used for Hosting Business?

Should I purchase an expired domain for my web hosting business or go for a new one?

if your business has a brand name (which I assume that it does) you should register a fresh domain name so you can register a brandable one.

otherwise, if you don't have / don't care about brands you can pick an expired domain to speed things up a little. @Nargil has great domain names if you need one from a trusted seller, but I don't think he has brandable domains that cater to your specific needs. But you can still check his inventory out...