Usually the problem is not the SID (apart from possible confusion) but the DBID.
If you have 2 (or more) databases with the same DBID you will face big problems i.e. maintaining a RMAN catalog database (the DBID is the primary key for the backup environment there).
It's also possible, that you run into troubles with Grid Control (I've never tried that)
You may, however, easily change the DBID (as well as the SID, if needed) with the Oracle "DBNEWID "