Sounds like you have an Axis or a Vegas unit. My speculation is that the "strange light" is actually the dome light. It comes on with the key and stays on a little bit after you shut off the key. I also think the odd blinking it does occasionally is either software controlled, or some sort of interference with another circuit.
On the left side the dash, by the lights and wiper switch is a rotating (up and down) rheostat knob on the dash that adjusts the brightness of the dash lights. If it is all the way up, that little light to the right of the radio will stay on. Rotate it back downwards.
X2, you feel a click on the dimmer wheel, up and down.
__________________ I'm wearing trifocals now but my hindsight is 20/20.
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Agree with those who mentions the dimmer switch for the dash lights. Turning it full in the one direction will turn on the LED "dome" light. You should feel the dimmer click into the position that turns on the light. Just back off the dimmer until it clicks.
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2015 Vegas 24.1 (E-350)