If you and the child have been TN residents for more than 6 months, you can file a name change petition in Shelby County Probate court. I'm not certain, but I would expect the Court to require a child support order in the Juvenile Court to be completed before approving the name change. You would then take you name change order to MS to change the birth certificate and other vital documents.
This is a fairly simple process if everyone is in agreement. Please contact me directly to discuss the private details of your case.