My iPhone 4 reboots by itself but only between 11pm and 12am

So, the phone is 2 weeks old. Brand new. It is not, nor has it even been jailbroken, nor have I tried to jailbreak it. The phone will work fine during the day. I use it as an ipod in my car. My stereo has ipod control which allows me to control the ipod through the stereo (change songs, browse albums, artist, etc).

So the issue started one night and I was confused. Then again the next night. That's when I began to wonder what was wrong with it. Browsed the net for like problems and the only thing I found that didn't require taking it in to have it worked on was an article stating that it could be the connector cable and the connection to the stereo. So I started trouble shooting. First, I plugged it in normally (USB connector to the phone), it rebooted. Then USB to the power outlet to charge, audio cable in the phone to the stereo for playback, rebooted again twice about 5 mins apart. Took the charger out, turned off ALL apps, just had the audio cable in the headphone jack, rebooted again twice. Got home, ended testing that night.

Drove to work the next day, plugged it in for the 35 minute drive to work, all's well. Head home that night and BOOM, problems starts up again. Left it off all cables, no apps running, just the phone itself sitting there, reboots every 5-9 minutes. all the way home.

So, go to work the next day, its fine, then at night, reboots. But I start noticing that its only doing it after 10pm and stops after 12am. When I wake up its fine all day, but at night its rebooting. Timed it yesterday and today, both times the phone was fine until 11pm. No restarts, reboots, or crashes. But after 11pm (now, because I'm thinking maybe time change- DST) it starts up with the funkiness.

I reading everywhere that its a hardware issue, but given the fact that it only occurs during the late hours and not constantly all day, wouldn't that make it sound less of a hardware issue and more of a software issue? Its running iOS5, again NEVER been jailbroken. I've restored the phone, restored to factory default, and don't want to try again as syncing takes WAY TO LONG.

Please if anyone has had the same issue and has come to a solution, lead me the way to resolve this. Thanks a bunch