There is no option to rename a table in BigQuery 😬.
You can copy a table in BigQuery, though. This doesn't incur any additional charges other than the additional cost of storage. Once copied, you can delete the original table so you only get charged for the storage once.
In the BigQuery UI, select the table you wish to copy, then push the
Copy Table button. Enter the desired new table name.
Or you can copy a table in the BigQuery command line tool:
bq cp mydataset.mytable mydataset2.mytable2
BigQuery documentation lists additional methods for copying a table (via API, with Python, PHP, etc). There does not appear to be a way to copy tables via a SQL command in BigQuery.