Both are acceptable: the reason at the time was x and it still is X, and will be in the future.
So which you use depends on the context.
If you’re explaining to your boss why you arrived late an hour ago, you would use is:
Gee, Mr. Sartorius, I'm sorry: the reason I was late is because I slept in this morning.
If at lunch you’re telling a friend what you said to your boss, you would use was:
So I told him the reason I was late was because I slept in this morning.
But if some weeks later you’re telling somebody why you got fired, you’d have to change the time reference:
So I told him the reason I was late was because I slept in that morning.
And by the way, you might not have been fired if you hadn’t said you slept in—that implies, at least in US usage, that you did it on purpose. Next time tell your new boss I overslept, and maybe he’ll be more forgiving!