If a product variant is renamed will customers still be notified?

As long as you edit the product variant, rather than creating a new variant, this will work fine. 

Back in Stock tracks product variants using the catalog id. This doesn't change when you update a product variant. It's fine to change things such as price, title, description and even the SKU. The catalog id won't change, so Back in Stock will continue to monitor the inventory level correctly.

However, if you are entering the product as a new variant under a different name a new id will be assigned to that variant. When this happens Back in Stock will continue to wait for updates to the old catalog id, and won't know you're now working with a new catalog id. If you do need to delete and re-create a product or variant simply drop us an email and we can link up your customer notifications to the new catalog id.