Rename a mongoDB database can be a difficult process because this operation will requiere a database outage, you have to make a backup of whole database, create the new database and restore into it. (mongodump, mongoimport) or by db.copyDatabase() command.
Those commands don´t take care of online operations, so, you need a database blackout to perform a database rename.
In Sepalo Software we have designed a procedure to overcome this issue, by using a mix of incremental database recovery and database rename, with only some seconds of outage.
The procedure is based in incremental backups (oplogs) and incremental restores into a new database (the renamed one) doing a previous oplogs entries substitution to rename namespaces.
Please, feel free to contact with Sepalo Software if you need more information.