I have a N8008 that I purchased new in 1990. It still works perfectly.
What you describe is normal operation of the viewfinder display illumination in a dark environment. The green light turns on with a half shutter press and blinks off for the exposure, then on again after the exposure.
The viewfinder illumination push button doesn't do anything in a dark environment. However, if it's bright enough that the light does not turn on automatically with a half shutter press then the button will turn the light on. The meter must be on first - half shutter press to display the meter info then press the illumination button. I don't know the reasoning behind this, but that's the way it works. Take the camera outside or aim it out the window during daylight, half press the shutter and then the illumination button - it should turn on.
You can cover the window with black tape if it's annoying. But then you will need to use the illumination button in bright environments in order to see the viewfinder info.