My Cinema Display blinks …3-4x during first the 5 minutes of use each day.  Is it dying?  A young 15" Macbook Pro drives it and I'm using latest Lion.
mDP?  Usually means hardware issue on the Pro - just replaced a user's 2009 MBP with a late 2011 and her otherwise-fine 27" LCD Display did the same thing.  AppleCare replaced the main logic board and no problem since.