OK, you shamed me into it. Get the
7809 datasheet. Get an LM7809, MC7809, whatever. Get a 0.33uF ceramic capacitor and a 0.1uF ceramic capacitor. You are going to be constructing the circuit in Fig. 8 of the datasheet. The pinout for the part is also in the datasheet.
If you enclose the circuit, put it in a metal box with the 7809 tab connected to the box by a machine screw, lock washer, and nut. This will dissipate the heat (about 1/2 watt max) to the ambient air. Make sure pins 1 and 3 do not touch the metal box. You will want to mount the 7809 before you make the solder connections.
I hope you know how to solder. If not, see . Solder one lead of the 0.33uF cap to pin 1 of the 7809, and the other lead of this cap to pin 2. Solder one lead of the 0.1uF cap to pin 3 and the other lead of this cap to pin 2. Keep the leads on the caps as short as possible. Solder the +12v wire from your cigarette lighter adapter to pin 1 of the 7809. Connect the minus wire to pin 2 of the 7809. The lengths of these wires is not critical.
Pin 3 of the 7809 is the +9v output to your headphone amp, and pin 2 is the other power connection.
Don't be confused by the rake-looking symbol (ground) connected to pin 2 in the schematic. This indicates the circuit common (GND) node, which will be taken care of when you hook all this stuff together as above.
I was going to tell you to put a fuse in line with the battery, but the lighter socket should already be fused in your fuse panel.
Let us know if it works.