I'm quite an amateur so I'm not sure what you meant by "Since this is a LC circuit, it may actually ring". How will the diode help?
You are mixing two things I wrote. Let me explain them separately.
The relay coil is an inductor. When you run current through it, the inductor stores energy. When you stop forcing current through it, it releases the energy by continuing the current flow. A capacitor also stores energy. In a LC circuit (L=inductance, C=Capacitance) you can have a situation where the capacitor transfers its energy to the inductor, which transfers it back to the capacitor, which transfers.... When this happens in resonance, you get an oscillator. Otherwise, the back and forth may happen a few times at decreasing amplitude before its stops. That is called ringing. Like hitting a bell and it keeps producing a sound for some time after you hit it. It is the same phenomenon in the mechanical world.
The diode is needed if you have just a battery and relay. When you disconnect the switch, the current in the relay coil wants to keep flowing (inductor releasing energy). This creates a condition called "flyback" which can damage the relay or the switch. To suppress the flyback voltage and provide a path for the current, you need the diode.
The reason you are getting double bounce is due to ringing. Another issue that you need to consider is that when you discharge the cap, the inductor is most probably making its output negative. If you have a polarized cap, it is not liking it one bit. The cap is going to fail a lot sooner than you are expecting.
The right way to do it is with a micro. You can either use a PicAxe or a straight PIC or one of about a dozen other micros and use it to control the relay. Or you can get a PIC or some other micro that has a USB port and make it look like a keyboard. I know that Microchip has reference design and reference code to do that for the PIC. I suspect so do all other micro manufacturers. You plug your device into the PC and the PC will think that you have installed a second keyboard and will accept keypresses from that keyboard. The down side is that you are looking at
A LOT more learning on your part. The up side is that once you have that figured out, there is
A LOT that you can do to enhance you cockpit (notice the greater emphasis on the second "a lot").
---55p