Here's one for the ABC's file:

Sounds like a bad positive battery connection either on the battery or starter, or possibly a bad starter solenoid. If your Accord is a low-miler I'd check the connections first. ANY corrosional buildup anywhere needs to be cleaned, then re-checked the next cold opportunity.

With any problem on an electrical system, I tell people to start with the ABC's Check the Alternator, Battery, Connections. 70% of the time you'll solve most electrical problems, and even some driveability problems, right there.

Connection problems tend to worsen with cold weather.