MySQL: Cloning a MySQL database on the same MySql instance

Using MySQL Utilities

The MySQL Utilities contain the nice tool mysqldbcopy which by default copies a DB including all related objects (“tables, views, triggers, events, procedures, functions, and database-level grants”) and data from one DB server to the same or to another DB server. There are lots of options available to customize what is actually copied.

So, to answer the OP’s question:

mysqldbcopy \
    --source=root:your_password@localhost \
    --destination=root:your_password@localhost \
    sitedb1:sitedb2

As the manual says in Copying Databases you can pipe the dump directly into the mysql client:

mysqldump db_name | mysql new_db_name

If you're using MyISAM you could copy the files, but I wouldn't recommend it. It's a bit dodgy.

Integrated from various good other answers

Both mysqldump and mysql commands accept options for setting connection details (and much more), like:

mysqldump -u <user name> --password=<pwd> <original db> | mysql -u <user name> -p <new db>

Also, if the new database is not existing yet, you have to create it beforehand (e.g. with echo "create database new_db_name" | mysql -u <dbuser> -p).