Owners can choose from Dead Rising 3, Forza Motorsport 5, Ryse: Son of Rome, and Zoo Tycoon. It seems that for the moment the ability to pick a free game is limited to those with faulty blu-ray drives, but it is possible that it would be extended to cover more types if issues, should other issues raise their head.
As for the number of Xbox One owners that are experiencing the Blu-ray drive issue, Microsoft claims that it is a very small number of Xbox One onwards who have had this issue.