Hi,
I just tried to reproduce the issue on my end but I'm not able to. When I copy a product without the name field filled in the variants, it's not filled either in the copy of the product. I tried to also copy the copy, and copy the products several times, but I'm still not able to see any issue.
So there must be something else to it.
Could you provide a screenshot of the main product and one variant before the copy, and same on the copy with the issue so that we can better understand the situation ?
If you want to just empty the name fields of the variants, you can do that with a mass action via the menu System>Mass actions. Just run a mass action with a filter on the product type to only affect the variants, and then use an action "update the values" on the product_name column without any value and it will empty the name of the variants.
Supposing that you also have Falang translations for the variant name (which I'm not sure that's the case?), then I'm afraid it will require the deletion via your PHPMyAdmin. Please understand that such support is not included even more so if it's for another extension. So we would have to bill you an extra for that operation. But first we need to make sure of what is your issue.