There are sites with tools that help you come up with fictitious names, including: wordlab.com, naming.net, wordoid.com, lingzini.com, namestation.com, some of these tools look up the domain name availability as you go.
Any name not already used in a .com web site name is probably not trademarked.
You can expect a totally non descriptive name (no web, no design in the name) might cost a bit more in branding promotion to get going.
Consider offering your services through freelance marketplaces like odesk.com, elance.com or vworker.com, you would be facing world wide competition so expect to make a strong grasp of English language and culture one of your assets. One design contest site that earns premium fees if you can submit a winning protorype is 99designs.com