Display turns on at midnight?
try to trigger it manually.

Root reqired:
Install BetterBatteryStats to view 'alarms', aka programmed low level device wakeups (you can find a free .apk on xda here, search for 'Attached' http://forum.xda-developers.com/showthread.php?t=1179809 ). set your clock just before midnight and reset bbs' logs.

If you can trigger it, you should see which app has caused a wakeup in BBS' log.