My suggestion is, simply list the device with the new brand name under the K number issued. If you already marketed the device under the previous brand name, the Registration and Listing website allows the use of more than one proprietary name for the same K number.
Also, you need to send a Note to File to the Office of Device Evaluations stating the K number and that your company changed the device name from the one on the original submission to the new brand name.
Please be noted that don’t forget to include the K number of your device on the envelope and send it to below FDA address
U.S. Food and Drug Administration
Center for Devices and Radiological Health
Document Mail Center - WO66 Room 0609
10903 New Hampshire Avenue
Silver Spring, MD 20993-0002
If the new brand name implies the claims or a use that was not cleared in the original 510(k) then you need to submit a new 510(k).
As MIREGMGR said, refer the labeling section of the FDA guidance on
Deciding When to Submit a 510(k) for a Change to an Existing Device. If you want to make any labeling changes keep details in your Design History File (DHF) and Device Master Record (DMR).