Choosing a business name for your website design business (or any business, really) can be a daunting task. You have to think of a name that's unique, memorable, and possibly relates to web design. Most importantly, you have to like the name, because you could be looking at it and hearing it for a very long time.
Step 1: Consider the Basics
There are a number of very basic fill-in-the-blank names which get the point across effectively, with no wordplay. They are evergreen (unless you change your name) and easily remembered. Consider names like _______'s Web Design, _______'s Website Design, ________ Designs, etc., where the blank is filled in by your name or nickname.
Step 2: Consider Meaningful Words
If you have any words or phrases which are especially meaningful to you, you might try to incorporate it into your business' name. Words like 'Love,' 'Peace,' 'Joy,' and the like tend to be overused, but more original words can be used. Think about your favorite flavors, smells, pets, colors, etc., to come up with a list of words that mean something to you. Make sure you come up with words that have a lasting meaning and that you will always think of fondly.
Step 3: Consider Songs
If you have a favorite song, glance through the lyrics and see if anything pops out at you. You don't want to use a section of lyrics that is too strongly connected with the song. For example, the James Taylor song "Fire and Rain" might inspire the name "Sunny Days Designs" or something along those lines.
Be careful when using song lyrics to inspire business names. On one hand, they can provide you with original and memorable names; on the other, you may find that you like the song less after some period of time.
Step 4: Make a List
While considering all of the above, write down potential names in a notebook or on your computer. Whatever occurs to you, write it down, no matter how silly you may find it a few seconds later. Come up with all the names you can.
Step 5: Narrow it Down
Go through your list and pick out your favorites - or at least eliminate your least favorites. Ideally, you should have around 5 names (or at least 2) which you could seriously consider as your business' name.
Step 6: Do a Check
After you've narrowed down your potential names, do a quick search on those names with Google to see if any websites already exist for a business by that name. Having an original name can help you eliminate competition, so narrow down your list further until you have a few names - or perhaps only one name - that isn't already in use. Also, consider the availability of the domain name you would want for your business. If you've gone with "Sunny Day Designs" as the name, you'll want to ensure that "sunnydaydesigns.com" isn't already registered. Use WHOIS.com to search domain names' availability.
After you've completed all these steps, you can pick the best name from your brainstorming list. Enjoy!