Our neanderthal kin had red hair and green eyes long before modern humans did, but the gene that gave them their fair colouring was a slightly different one from ours.
Red hair, like blonde hair, is thought to have developed (along with light skin) as a response to sunless climates. People with these traits were able to get enough vitamin D from the sun, which was difficult for darker-complexioned individuals in rarely sunny environs.
Source: Flickr User mueritz