Most likely corrosion in the wires/connector. If you have 12v on 1 side of the connector and 3v on the other then there is a resistance between them. You can set most multi-meters to read resistance (ohms, or the little upsidedown horseshoe looking thing ). I would put 1 meter lead on each side of the connector and see what it reads. Under normal conditions this would be a very low number probably less than 1 ohm. Based on the numbers you gave, you will see a much higher number.
You might be lucky and it's just grime or corrosion in the connector itself. In which case you can probably clean it and coat it with dielectric grease to prevent it from coming back. If it's in the wire, then you're going to be replacing the wire. I would also measure the connector end of the wire and the output end (which ever bulb it goes to) for resistance. Again it should be a very low number...high number is bad.