In a 2013 study from the International Journal of Medicinal and Aromatic Plants, flax was shown to have a significant positive impact on hair thickness. This was the case whether the seed (flaxseed) was ingested or the oil (flaxseed oil, or linseed oil) was applied topically.
The study was performed on rabbits in two separate trials -- oil application vs seed intake. Hair width increase via the application of flaxseed oil to the scalp was detected after four weeks while that of flaxseed ingestion was observed in the twelfth week. Hence, flaxseed oil demonstrated faster results. (It also had more of an impact.)
Further research is necessary to determine the mechanism by which this hair thickening occurs.
So what is the verdict? Try it out and see how it works for you! Do your own little experiment.
Beroual, K.; Maameri, Z.; Halmi, S.; et al. "Effects of Linum usitatissimum L. ingestion and oil topical application on hair growth in rabbit." International Journal of Medicinal and Aromatic Plants. Vol. 3. Iss. 4. pp. 459-463, 2013.