... You'd want to discuss that with a business and tax attorney when the time comes to move. ... If you already know the name you want to use and the state to which you will move you might want to search to see if your chosen name is already being used for a similar kind of business in any place you hope to do business. If you pick a name that is really unique you have less risk you'll have a problem with trademark infringement issues later.
Taxing Matters brings up a good point. You can register your business name with a state if no other business is using the same name but you may not be able to use that name in commerce as an identifier for your business product or service if the name is already someone else's trademark.
You will want to do a trademark search before selecting a name for your business.
You can start your search by looking for federally registered trademarks at the United States Patent and Trademark Office -
https://www.uspto.gov - and expand your search to state registered marks and, finally, look for unregistered but in-use marks (because trademarks in the U.S. do not need to be registered to be rights-protected).
This can be a time-consuming task so it can be easiest to create for your business a unique name (a made-up word, two unrelated words tacked together, a word used in a way that is different than its dictionary meaning).
I agree with Taxing Matters that discussing your plans with law and tax professionals is wise and advised.
Good luck.