You generally don't want to use the product endpoint to make changes to product variants (option) as it tends to be destructive. For example, when updating a product's variants, if you don't include the IDs of every other variant that you're not updating those variants will be deleted! So, if you find yourself using the product endpoint to make changes to variants and something "isn't working", it's probably to save you the headache of accidentally destroying your work.
That's some more general advice but to answer your question you would want to use either the REST Product Variant resource or the GraphQL productVariantUpdate mutation. The REST page includes an example of changing the name of an option (the PUT example).