If your OEM handles everything (manufacturing, packaging, labeling) EXCEPT distribution to end users (i.e., you are using their 510(k) and function merely as a private-label brand of their product), then you can leave their name off and only say "Distributed By" [your company name].
Since you are repackaging/relabeling the above situation does not apply. You are a repackager/relabeler. You are not a full-blown manufacturer (i.e., you don't need Design Controls and such).
Regarding the 510(k), here's what the FDA has to say:
When a 510(k) is Not Required
[...]
4. In most cases, if you are a repackager or a relabeler and the existing labeling or condition of the device is not significantly changed. The labeling should be consistent with the labeling submitted in the 510(k) with the same indications for use and warnings and contraindications.
Earlier on in the document it states:
The following four categories of parties must submit a 510(k) to the FDA:
[...]
3. Repackers or relabelers who make labeling changes or whose operations significantly affect the device.
Repackagers or relabelers may be required to submit a 510(k) if they significantly change the labeling or otherwise affect any condition of the device. Significant labeling changes may include modification of manuals, such as adding a new intended use, deleting or adding warnings, contraindications, etc. Operations, such as sterilization, could alter the condition of the device. However, most repackagers or relabelers are not required to submit a 510(k).
Reference:
fda.gov/MedicalDevices/DeviceRegulationandGuidance/HowtoMarketYourDevice/PremarketSubmissions/PremarketNotification510k/default.htm
That said, you are likely OK using their 510(k), registering as a repackager, and building out your QMS accordingly.
Thanks,
~Josh
