SQL Server 2012 Restore backup to new DB Name
Based loosely on Example E in the documentation, open a new query window and run:

RESTORE DATABASE MyDB_Copy FROM DISK = 'C:\blahblah\MyDB.bak'
  WITH REPLACE, RECOVERY,
  MOVE 'MyDB' TO 'C:\blahblah\Data\MyDB_Copy.mdf',
---------------------------------------^^^^^
  MOVE 'MyDB_log' TO 'C:\blahblah\Data\MyDB_Copy.ldf';
-------------------------------------------^^^^^


The logical names are not important; the physical file names are. This makes assumptions about your logical file names and that there are only two; run EXEC MyDB..sp_helpfile; to be sure.

If you need to restore logs, then change RECOVERY to NORECOVERY:

  WITH REPLACE, NORECOVERY,
----------------^^


Then you can issue a series of:

RESTORE LOG MyDB_Copy FROM DISK = 'C:\blahblah\file1.trn' WITH NORECOVERY;


And on the very last one:

RESTORE LOG MyDB_Copy FROM DISK = 'C:\blahblah\fileN.trn' WITH RECOVERY;


Or if you only need part of a log up until a point in time (I assume you've checked where the LSNs and times are so you know exactly which files you need):

RESTORE LOG MyDB_Copy FROM DISK = 'C:\blahblah\fileN.trn' WITH 
  STOPAT = '&lt;some point in time Friday&gt;', RECOVERY;


The way you said worked in previous versions would never have worked, unless the backup came from a different server. By default it will try to put the new mdf and ldf files in the exact same place, and this isn't possible.