So I recently had a no-car-start problem and narrowed it down to the relay. I hooked up a voltmeter to the power going into the relay and it was reading ~12.1-2 volts so then I took apart the relay and manually touched the two copper pieces together, car started right up!

I was looking at the relay to try and diagnose what could be wrong with it, and the only thing I could see being an issue was this little piece that sits between connections?

What is this piece called and co0uld it be the culprit to a bad relay?

 

That is a resistor, a 1k Ohms resistor (Brown, Black, Red, Silver) to be exact.

Of course, if that was part of the circuit and not connected, the relay would not work properly.

The resistor doesn't look burnt, you can test the resistance with an ohmmeter to see if it is within specs. If it is within specs., you solder in back on where is was disconnected by bridging the resistor's leads or desolder the old resistor lead remnants off and replace it with a new resistor.

You do not want to just bridge the connection with a straight piece of wire or copper like you did, as a resistor is used to reduce the current flow and voltage.