How do I add databases from a crashed MySQL 5.1 server to a new MySQL 5.6 server
I have resolved this issue using the instructions posted here: http://dev.mysql.com/doc/refman/5.6/en/innodb-backup.html

I was unaware that by default all data in InnoDB tables across all databases on the server are stored in one single file at the root of the data directory: ibdata1.  That file, and I believe the log files, must also be copied along with the folders containing the FRM files (which apparently just store the table structure).  The side effect of this, of course, is that any InnoDB tables already defined on your new server will be rendered unusable.