If you use a long business name, try and find a good shorter domain name for the business that's not a pain for customers to type (or to remember).
For customers who search, most of the time they will probably remember a word or two and Google you up. The first time they search they might browse a few sites and bookmark a few, sometimes not. So it's a good idea to have a memorable name, but shorter names are generally better. If you have a long business name, try to make an interesting acronym out of it. Or if you have a name like Happy Turtle Web Design, be sure you can register happyturtle (.com/.something, avoid the new TLDs for now). Even a hyphen in the domain name is better than nothing, so if you find somebody squatting on happyturtle dot something, doing not much with it at the moment (if anything), add a hyphen like happy-turtle and use that.
No need to try and game Google here, not that it's an easy task, just make it plain that you're in Web design, try not to wander off topic, keep your keywords reasonably close together in paragraphs describing the business, have standard pages like "About Us", "Contact", portfolio or examples, and boost your rank a little by forcing https (SSL/TLS) connections. There are good free SSL certs you can request, for example startssl.com.
If you host in the cloud yourself, e.g. with AWS or Azure (quite easy to pick a Web site configuration pre-rolled from here), you may like to add a little protection for DNS and traffic with CloudFlare and request a free SSL cert from them. GoDaddy have cheap SSL certificates if you want a cheap EV SSL (not necessary to begin with, just domain validation is fine), although to get a good grade (e.g. check with ssllabs.com) you have to configure the server to junk weak ciphers and disallow downgrading the cipher and/or protocol (e.g. to SSL from TLS).
I haven't voted or suggested a name, since you should pick the one you like. Your business, your name, your rules.
Once an interviewer suggested that The Beatles were so successful because they were called The Beatles, and what was in that name anyway? Both John and Paul responded that it's a name for something, like shoe. Paul agreed, saying they could have been called The Shoes. Doesn't matter that much. Be good at what you do and if you can't play in the biggest market, choose a niche market and try to cover it as well as possible. Reputation is vitally important for small business. Other small businesses are likely to talk.
I hope to have helped in a small way by suggesting more careful thought beforehand to help get up and moving more quickly and easily. Most small businesses will fail, and it's good to do a bit of research, check out your competition, etc. I know it's obvious and the basic stats are out there everywhere, but it is an important step. Then you can remove sheer luck and other uncountable/unknown bits from the equation to improve your chances of success.
So, Pink Pander or Happy Turtle or something else? Anything, so long as it represents something and does it well. All the best!