There is no clean easy way but it is certainly possible:
Create a Dummy entity (new_dummy) with just a primary attribute (new_name)
Add a dataset to any reports you want to track that does a query for new_dummy where new_dummy.new_name = ReportName
Create a RetrieveMultiple plugin that looks for new_dummy queries and retrieve the new_name value from the query. How you store this information is your choice.
Now you have a plug-in that is triggered on each report execution and has the name of the report.