There are many sites that generate random names (for businesses or other things), so many that I've stopped bookmarking them. But while they might loosen up your mind for a brainstorming session, if you really want to name a new business, I doubt you want a name chosen at random.
As for doing a search on the availability of a name, there are several places you'll need to do that, depending on where you plan to do business. I'd start with the USPTO trademark database, because if someone else has trademarked the name you're thinking of, you have to start over and come up with another one. But you'll also have to check the corporate/LLC databases for your state and the fictitious business name database for your county.
Dot-o-mater is a great website for generating a business name. It's primarily geared toward Web 2.0 companies, but I find it can be used for anything. It has a simple mode of just generating pseudo-random names and a more sophisticated, and I find more useful, mode of combining different permutations of prefixes and suffixes to generate unique names. It also has a very useful feature to bulk check which domain names from the list are unregistered.
The orignial random name generator for businesses is also quite useful. I doubt it checks the business name for matches to domain names, but still, it is easy to use and will inspire some ideas ......
Random Business Name Generator
It has a nice touch of giving you the option of business you want to start, matching the result to how companies in that field are typically named.
However, if the aim is also to have lots of Traffic at your business website, it might be more productive to start with brainstorming your business ideas. Pick the best concept. Find the most profitable keywords ..... and incorporate the best keyword(s) as part of your domain name.
There are +56 million websites out there. Without proper planning, you'll be lost in the multitudes. With proper planning (and it does not have to be a costly exercise) you can be in the top 10% of websites. A proper keyword-rich domain name gives an added boost.
Going about it as most people do costs not only money and dismal results, but also time lost. Lost time cannot be recovered or recycled.
So .... although the above may give you some ideas. There's no substitute for a little old fashioned brainstorming and creativity. All depends on what your purpose is for the name you finally choose.