How do cats purr?

Cats have a special wiring! The wiring travels from the brain to the muscles in the voice box, and this wiring is able to vibrate the muscles so that they act as a valve for air flowing past the voice box. The muscles work both during inhalation and exhalation, which creates the impression that cats can purr continuously. The air passes through the valve, which opens and closes rapidly to create the purring sound.

There are two explanations for this, Tivoli gave one and the other is that the purring sound comes from the blood vessels. So actually your friend is right, no one knows for sure, but it's not a complete mystery.

One theory is that cats produce the purring noise by vibrating their larynx, or voice box, in a particular manner. A timing mechanism in the brain sends neural messages to a muscle in the larynx, rhythmically opening and closing the air passage approximately 25 times per second.Combined with the steady inhalation and exhalation of air as the cat breathes, a purring noise is produced with strong harmonics.

One theory held that purring involved blood hitting the aorta. Another possibility was that another area of soft tissue or muscular tissue in the neck or torso (e.g., the diaphragm) similarly vibrates. Another held that purring might have been caused by the vibrations of the hyoid apparatus, a series of small bones connecting the skull and the larynx that nominally serves to support the tongue. Yet another theory held that cats might possess a special purring organ, though none was found.

 http://en.wikipedia.org/wiki/Purr#How_felines_purr