A few ideas - Check your protection and scheduling
The IPAQ turns on at midnight to perform some administrative functions - you can't prevent this from happening. What kind of setting did you have for it to turn off on battery power? Had you left anything like Windows Media Player running or WIFI or Bluetooth when you turned it off? If so, when it came on at midnight, it might have detected activity and stayed on (this will happen if you left music playing and then turned off the IPAQ).
It could be your email trying to connect to the internet every 5 minutes or so.
It could also be a number of pieces of software, any weather software trying to do an auto update.
Just look at the settings of your software and see if any are trying to connect automaically.
May seem like a simple thing, but mine stopped turning on spontaneously when I turned the HP Protection tools OFF. While on, the 4705 seemed to come to life at the same intervals that it was set to lock the system. Now that I turned it off, it hasn't happened since.