Radon is a product of the decay of naturally occuring uranium in the earth, and it's estimated to be responsible for perhaps 2-3% of lung cancers overall. In one reference (here) doesn't offer any details of where the numbers come from, an estimated 26% of lung cancer in never-smokers is caused by radon:

(Click on image to enlarge)

In addition to radon gas, which is inert and is not felt to contribute to cancer-inducing effects, radon breakdown products are called radon daughters, which are radioactive metals that attach to air particulates. These may potentially cause lung cancer when inhaled, where they can go through the pulmonary system and land on cells in in the lungs, inducing damage to the DNA of these cells. We know that underground miners with a significant occupational exposure to radon have a markedly elevated risk of developing lung cancer that is in the same range as that of long-term smoking (abstract here, another here). In addition, radon can accumulate in indoor spaces, so exposures can occur in homes or offices, and it's estimated that indoor exposure to radon accounts for 50-80% of the total amount of radiation exposure a person receives in the US (from the epidemiology chapter of this book). However, it's hard to determine the degree of risk from such exposures. The amount of exposure in a normal lifespan is considerably less than that of working as an underground miner, but there doesn't appear to be a "danger threshold". Instead, it appears that the risk of lung cancer increases gradually along with increasing radon exposure.

There are a few other non-tobacco risk factors I'll also cover soon.