Tim & Viper -
I know what you are saying, but given that the horn isn't used continuously(except as noted in jokes above), I don't think it's worth it.
In theory, you're correct, tho.
Actually, besides the possibility of overheating the wires (yes, it's not really that much of a problem), using a larger gauge wire and fuse will allow the air horn to draw the maximum power. Air horns use a small air compressors to create the air flow. More power, and the compressor runs stronger and faster. The result is a louder horn, and a longer lasting horn.
It was common for people to add an air horn to early Miatas and use the stock wiring with a larger fuse. But if you ran a higher current circuit triggered by a relay, you ended up with a louder horn and eliminated the possibility of blowing a fuse. I ran my Miata with a bigger fuse for about a year with no problems, but when I rewired things for a dedicated circuit and relay, the horn got louder...
The wiring that you want to use is shown in my crude drawing below. Basically you use the stock Purple with Yellow tracer horn wire to trigger the relay. Connect the wire to pin 85 or 86 of the standard relay - this is the relay coil circuit. Connect the other pin (86 or 85 - which ever you didn't plug the other wire into) to ground. This can be the factory horn's ground wire, a bolt on the chassis, or the battery negative post (on the Exige).
Then you connect a battery positive source (the battery post on the Exige or some other source on an Elise - some tap into the power under the dash, or the power window 20 amp circuit) to pin 87 on the relay. Connect pin 30 of the relay to the air horn's positive connector. Connect the air horn's negative connector to ground, or the battery negative post on the Exige. Don't connect it to the original horn ground wire - just as the positive wire needs to be big enough, so does the ground.
On other cars I've done this wiring to, I mount the relay up near the other fuses/relays in the bay. I tap into the power, add an in-line fuse, and connect to the relay. Then I run three wires down to the horn area. Two are naturally the horn Positive and Negative wires which I connect to pin 30 and ground at the relay end. The third wire is used to plug into the OEM horn wire, and brings the "signal" back up to the relay (connected to pin 85). I connect pin 86 to the same local ground that I used for the horn's negative wire that I ran.
Doing it this way does the least harm to the OEM wiring - it lets you remove the modification and return things to stock if you should ever need to. Mounting the relay up by the fuse box also keeps it up out of the road grime and water. "Physically" you could mount the relay in front with the horn, but "electrically" it doesn't matter where it's mounted - the more protected, the better.
Incidentally, I always use a socket for the relay. Just using spade connectors on a relay is an easy way to mess up when you are plugging things together. It's also harder to diagnose a loose/dirty connection farther down the road. They sell relay sockets with "pigtail" wires already connected, but it's far better if you find one with the loose insert-able spade connectors and crimp them onto your own wires. This eliminates having to splice together extra wries, and makes for a much more professional installation.
All that said swapping the fuse and "overloading" the wiring works, it just that supplying the proper circuit and fuses works better. How much better? That depends on how much work you want to do...
Here's my crude wiring diagram. Below that is the pin locations on a standard automotive relay (be careful, there are a few variations on this that are slightly different - but the standard is shown).