There's no claim here. When your mom bought the house, she agreed to the terms of the HOA... if the HOA says in their contract that you may not operate a home business within your home, then she is in breach of contract by doing so. If there wasn't a clause about this in the HOA contract, then they obviously wouldn't be bothering her about this.
No, the state is NOT violating your rights or your mother's as an American - they are not "hindering" her ability to earn a living. You have to follow rules like everyone else. If she wants to operate a home business, then she needs to purchase a home with no HOA or an HOA that will allow her to do that.
The reason why some HOA's prohibit this is because it is a residential area where people live, raise their kids and walk their dogs. It is not designated to be an area of business. There are associated risks that come with operating a business, and if something goes wrong with it then it is possible that the HOA could be liable as well [[for allowing her to run the business]]... and they don't want that liability. Even if it's something as harmless as a flower business of some sort - no one could possibly get hurt doing that. But, if the HOA let someone operate that business in their home, to be fair they would also have to allow your crazy neighbor to operate his chainsaw-juggling-class business as well. Does that make sense?
And a final note - you as a minor cannot file a lawsuit. You parent/legal guardian would have to do it on behalf of you. And in this matter [[lets just pretend there IS a legitimate case here and you are of the age of capacity]], you don't have any business in this matter anyway, therefore you would not be able to participate as a party in this "lawsuit."
[[That means I work with lawyers and assist them with cases and legal research]]