Teaching Dilemmas Series: Survey Question

I am going to be starting a series of posts on the ethics of teaching and ethical teaching dilemmas. To get things started, I wanted to present an ethical dilemma that is very familiar to me (and I'm sure, other educators). I have no illusions about the number of readers I have or the likelihood of engaging with this poll, but I still thought it would be fun to post. Perhaps in a few years, I'll have a consensus ;)

Here is the scenario: You are an instructor at a large university in Canada. You teach one of the larger second year courses in Chemistry. One of your students has come to office hours to ask for an exception. The student had completed the homework that was due yesterday and claims the homework was actually completed well in advance, but the student forgot to turn it in yesterday between classes because he was preoccupied by a recent breakup. Because the homework assignments are worth so little (only 5% per assignment), your course policy is that late submissions are not accepted. You don't know it, but there is another student in the class with a similar situation, except it was the death of his dog rather than a break-up. That student has decided to accept the consequences of his actions, and will not approach you to ask for an exception. The student in your office is one that you know very well, and you believe his story about the break-up. He is asking you to make an exception to this rule and allow him to turn in his assignment without penalty.