Your friend probably either was the victim of a scam or heard of an anecdotal story and misunderstood.
Regular cell damage due to bruising, local bleeding and other processes like inflammation can sometimes lead to cancer if the irritation is chronic. There are also a few viruses (like HPV) that can lead to changes in cell structure and eventually cancer.
Now, how would cat hair cause such a chronic irritation? My honest oppinion is that it can't. The usual hairs in a cat's coat are about as thick as human hair and our bodies are well equiped to deal with those if not ingested in vast amounts.
Whiskers are much thicker, though, and the scaly outer structure gives them interesting physical properties. If you let a hair glide through two of your fingers, you'll notice that it glides very smoothly from root to tip, but with much more resistance from tip to root (the same applies to human hair, but to a lesser extent). This means that if you ingest a piece of whisker, if it survives all the way into the intestine and if it happens to get caught root first in your intestine (or you manage to poke it into your skin somehow), it can only move in one direction. It migrates very slowly through the tissue, causing constant irritation and microscoping tissue damage, and the body will eventually encapsulate it in a dense ball of cells (commonly known as "tumor") as a protective measure. This process is known from awns or "mean seeds" (the tough hairs at the tips of grass seeds) as documented here.
To be honest, I have no idea if this can actually happen with hairs. The human body is able to digest hairs in the stomach and might be able to desintegrate cat hairs outside of the digestive system. But I couldn't find any evidence that any human ever developed cancer due to cat hair.