As a private chef you should brand yourself based on your personal style as much as your cooking style. In what area of the country are you going to be cooking? What is your demographic? I just spoke to two owners that were not having the results they expected and I pointed out their store names as part of the reason (layout and selection were others). Also, they created what they wanted on the menu vs what the demos indicated. There was another Chef/owner I spoke with that had great success polling supermarket shoppers in the area where he opened his shop. He went in both He put a slight twist on his dishes to make them interesting and has lines out the door and runners to keep up with his customers.
The point I am making, if you have not yet you might do a bit of research to support the foundation of your business, and in naming it consider a logical approach to your direction longer term to establish your reputation, and spice it up enough to make people that see your name go hmmm...that should be your personal brand recognition as well as the brand you are creating.
Ever read the "The Tipping Point"? More businesses fail due to lack of name recognition than the products they sell. All points in your business should connect. A bit Zen, but the percent of successes indicate that to be true.
One last question. Will your name tell a story? One to share? Consider it.
Your name is as important (and personal as any part of your business), as it should state your style, direction, level of quality, and those intangibles, creativity and confidence. Good luck!
TK