I had great success with the drop the lowest exam score method (which allowed people to miss one and meant, if they didn't miss any and did well early, they could skip the final). On the first day, I made it clear that car problems, oversleeping, etc. would not change that policy. I did make exceptions for a few students that had documented (doctor's excuse, police report, obit) issues though. Students were fine with it and I never had any issue with enforcing the policy.
On a related note: I had a student turn in a paper on the day after his mother died, which was the due date (I totally would've made an exception for him!), so now all other excuses pale in comparison... If he did it...