Say I named my Ditto BLOBBY. If the opponent (say an annoying sibling) wasn't paying attention until the attack was over, would the copied Pokemon have the name "BLOBBY" or the species name of the other Pokemon, or the nickname of the other Pokemon?
The Ditto will keep its nickname when it transform. So if you named it Ditto Blobby it will still be Ditto Blobby.
Source-Experience