I believe those are called ghost lights. I had that happen months ago. It was there one day, and gone the next. At the time, I had the popular wiring ground mod (perform a search for it here if you don't know what it is). I removed the wires for that mod, and never had the problem again.

You could try removing/adding the wire ground mod, depending on what you have now. I would highly suggest that you buy a voltmeter, though. They're pretty cheap — maybe $10 - 15. Test the voltage on your battery when the car is off. It should read 12 - 12.5 volts. Anything less, and your battery is probably bad. Test your battery with the car on, and it should read 14 - 14.5 volts from the alternator charging it. Anything less, and your alternator has a bad hookup, your alternator is shot, or maybe your battery and alternator are shot.

Something that someone just told me recently is that batteries and alternators often die as a pair. The battery starts to lose its juice, so the alternator has to work near its full capacity in order to keep the battery charged. Unfortunately, alternators can't work at full capacity for very long.

So, you replace your alternator, but it dies again soon because the battery is still losing juice.

Either that, or you replace your battery, and it dies again soon because the alternator is on its way out. Then you're back to square-one, cause your alternator is still on its way out.

Anyway, point being, you need to make sure your battery is ok before you do anymore driving on that new alternator. Even though it's new, it could die from trying to keep a bad battery charged.