Can I change the SID of an Oracle database?
You need to recreate the control file

This post by Kaunain Ahmed describes the necessary steps:


  
  do: alter database backup controlfile to trace;
  extract the "create controlfile" command from the
  background-dump-destination tracefile.
  shutdown the DB.
  Change the DB-Name in your init.ora and change the init.ora
  Change the SID in the /etc/oratab or /var/opt/oracle/oratab
  Change the SID in your environment and source it
  Startup the database to mount-status
  startup mount
  Re-Create the controlfile with the statement from position 2.
  Do a alter database rename global_name to 
  10.Change the TNS-Configuration accordingly
  $ORACLE_HOME/network/admin/*.ora Look for SID and GLOBAL_NAME 
  


There are other tools referenced in the thread.

Here's a post by AskTom which references the process in more detail. While it's for 10g, it should still work.