Researchers at the University of Louisville decided to test the anecdote directly, since red hair comes from specific mutations in a receptor that also sits in pain pathways in the brain.
They studied ten naturally red-haired and ten dark-haired women, raising the gas anesthetic desflurane until each stopped responding to a painful electrical stimulus. Redheads needed a concentration of 6.2 percent on average, compared with 5.2 percent in the dark-haired group, about 20 percent more.
Blood tests showed nine of the ten redheads carried mutations in the melanocortin-1 receptor gene, the same gene responsible for red hair and fair skin. The same receptor family appears to shape pain signaling, which may explain the higher dose. The finding has since nudged some anesthesiologists to plan for extra medication in red-haired patients, though it has not changed formal dosing guidelines.

Read the original study
Liem et al., Anesthesiology, 2004 · doi.org/10.1097/00000542-200408000-00006
A hair color that changes a hospital dose.
Source
- Liem, E. B., Lin, C.-M., Suleman, M.-I., Doufas, A. G., Gregg, R. G., Veauthier, J. M., Loyd, G., & Sessler, D. I. (2004). Anesthetic requirement is increased in redheads. Anesthesiology, 101(2), 279–83. https://doi.org/10.1097/00000542-200408000-00006