Decide your business structure
Oftentimes, many people decide on a name after much consideration, but then they realize that the name does not completely meet the requirements of the business structure that they have decided to choose. For instance, if you decide to create an LLC after picking out a name, you may soon realize that in most states you're required to actually have the words 'limited liability company' or some sort of abbreviation of this, or that you've used a word you're not allowed to, which can result in you having to start straight from square one.
Choosing a business structure before can also be helpful as it means you know what type of business you want to do - whether it's going to be formal or more informal, and so whether you have more leeway to have a relaxed name or not.
The name needs to sound good
While you may think your name is great because it meets all requirements, is not too long and sounds like something other businesses like you would use, you may find it's pretty ordinary and maybe even boring. You want a name that rolls off the tongue, a name that grabs attention and a name that's going to be in someone's head for a while.
It might help to say the name aloud so you get a feel for how good it sounds, and even better to get someone else's opinion - after all, if you've been working on a name for a few hours you might come to a point where every name sounds great.
While you do want your name to sound amazing, you also want one that summarizes what you do in a few short words very well, or at least one that alludes to what you are selling with your business. Don't try too much to be too much with your name - show your niche, and sell that, because that is what you are best at. That's what makes you special and why a customer should pick you over another business.
You also want a name that means something. It has to have specifics.
Test your name out on Google AdWords
AdWords has a great feature - a 'find keywords' tool, that will list similar search phrases along with how many global and local monthly searches each phrase is getting. So if you do a search on Google AdWords with the name you're considering or words related to that, you can find how popular searches for things related to your name are, as well as finding out if there is a slightly different name that can get more attention for you from the Internet.
If you want to get really advanced you can try to use a name that can even eventually be used as a verb, or give many words related, as if creating your own 'language'. For instance people who go to TED conferences often call themselves 'TEDsters'.
Make sure you can trademark the name
Often when people are starting a business they don't plan very ahead with how big they are going to make it, or what they will do if it starts getting successful and they want to branch out more in order to get more clients. A huge part of planning ahead includes the name. You might not want a website in the beginning, or to trademark your name because you do not think it will be that important. However, when picking a name, sometimes it's useful to check whether it is available as a domain for a website, and trademarking just in case your business booms. This means that even if you did not want it before, in the future you can make a website if needed.
If you're ever lost in picking a business name, sometimes it's very useful to use a company name generator, such as TRUiC's name generator, which brings up many names in a matter of mere seconds.