clean power, no voltage drop
The idea is to give clean, full power to a device, usually the high current items like headlights, horns and clean power to the audio devices. The wiring is a compromise, the smaller (read lighter) wires have more resistance than large wires and so the voltage will drop if you have to run very far. That is why you see large wires going to custom stereo installs.
Electric motors run hot and will burn out if not supplied with voltages close to their specification. So using the existing horn wires to drive a compressor type horn (Steibel) can be considered a bad idea. You will not provide a full 14.4 volts to the horn as the small wire will drop the voltage. Same goes for HID and other high end head lights.
One way to up the voltage is to run a large cross section wire from the battery or other primary terminal to the device and turn the current on with a relay controlled by the existing wire. You can use the existing horn wire to close the relay that turns on the large wire horn supply. Most horn instructions include a wiring diagram and often the relay itself.
My suggestion was to include a large diameter (small gauge) wire from your battery to the front of the car. The Exige also has a terminal at the front, under the left access panel, to be used for jumping or to provide power so the doors can be opened. (The engine hatch cannot be opened from outside the car, so it is important to be able to open the doors...) I obtained a power source for my stereo from this junction block.
I made the decision to use the existing wiring to drive my horn. Many will tell you I am living dangerously and putting my car at risk of a fire. However I decided the horn is used rarely if at all and only for short durations. You should not do anything to your car unless you know what you are doing, and any other disclaimers you want to put here...