As an applicant who lives in NYC and has already submitted an application for this round, I think a day (or more) extension would be helpful. One day should be enough for those who lost time preparing for the storm but were not affected enough to drop their whole application(for this round) due to significant loss.
I disagree with the suggestions to add a "I'm late because" or special consideration for late applications. This will postpone a decision and create unneeded complications with the selection process. What I can suggest, is that YC creates an "I will be late on my application because..." area and provide a 3 day extension for those groups. At the very least, this gives YC an opportunity to see if anything is coming in and allow for YC to make a decisions with some "wiggle room" for the late applications. I'm not sure how the selection process works at YC, but the bottom 2-3 fundable on-time applications could be on "stand by" until all applications are submitted and a final decision could be made for the last 2-3 groups.