There's a reason puppies are cute--it's to help owners put up with all the early morning "I want to go outside" whining. Puppies don't want to dirty their surroundings anymore than you want to stay in nasty dirty gym clothes for hours.
So just be persistent about taking the little darling outside (yes, even at 2 am).
Accidents do happen, they're inevitable. Be they owners deciding to ignore the puppy sitting staring at the yard intently, or the puppy waking up and getting too excited. Just move on and clean up the mess and don't forget to spray a de-odorizer/anti-stain spray. The de-odorizer spray makes the scent go away so the dog (or another dog for that matter) doesn't smell it and want to mark in the same spot again (beside the obvious that we don't want to smell it either).
As for that precious hardwood floor we discussed earlier, there's hope for getting the odor out of it as well: eHow