The best place to start the process of buying an asset of another company (be it the name, customer list, inventory or any other item) is with an attorney. You'll want to be very clear about what you're buying--is it just the name, or are there obligations (such as debts) that go along with it? Have you done your due diligence to make sure there's no "bad will" connected with the name?
Also, does your uncle intend to stay in business? If so, he’ll have to change the name of his company. If your uncle has trademarks or other intellectual property connected with the name, you’ll want to address that, too, in a written agreement.
Once you have worked out the terms, only then do you want to file any necessary papers with your state’s secretary of state. All the more reason to start with an attorney, who will identify in advance the issues you need to address.