How can I migrate data when Migration Assistant doesn't recognize Time Machine external drive?

I'm trying to migrate my files from my time machine backup (external drive 500 Gb) to my MacBook Pro with Lion 10.7.1. It has almost empty brand new internal hard drive 500 Gb.

I can see the external hard drive in the desktop (is connected to my MacBook Pro with a FireWire cable), but the Migration Assistant is unable to recognize it!

What can I do to migrate my files?

Have you tried right clicking on the time machine icon on your dock, and telling it to browse other time machine disks? See if Time Machine will recognize it as a backup, and see if it will restore any files that way.

When you select Restore From Time Machine backup it doesn't even see the external drive or list it/any backups?

Boot to your recovery partition holding down cmd+R on restart. That will give you the option to restore from time machine backup. That will definitely find your time machine backup