DIAGNOSIS This is usually a notification or a scheduled routine triggering the screen, or the ambient light sensor adjusting.
FIX Set up a Do Not Disturb schedule. In the Alexa app, go to Device Settings > Do Not Disturb and set hours (like 11pm to 7am). This silences notifications and keeps the screen dark. Also, on the device, swipe down and make sure "Auto-Brightness" is turned off, as it can react to shadows.
IF THAT DIDN'T WORK Check your Routines in the Alexa app. You might have one scheduled (like a "Good Morning" routine) that's set to run at a weird time and is turning the screen on. Also, disable "Continue Conversation" in Settings, as it can wake the screen listening for follow-ups.
HEADS UP If you use it as a clock, go to Home & Clock in settings and make sure the clock dims appropriately at night. You can also turn the screen off completely by saying "Alexa, turn off the screen."