Color naming is an active field of research and has been for decades. See this example of work by the US National Bureau of Standards (NBS) from 1965. I include this reference only because it shows the length of time people have been thinking about this and that technology keeps making the problem fresh - notice the reference to reproducing the colors in glossy paint (instead of on a computer monitor).
That publication is an extension of earlier work by the NBS and the Inter-Society Color Council (ISCC) created a method for naming colors.
Since that time, the color naming system has been applied to naming lights, medicines, and dental porcelains, among other things.
The ISCC-NBS color naming system is not the only system. Munsell is the most popular commercial system in the US. It has been used for naming colors in a variety of situation, e.g., soil colors. Crayola has their own color naming system. As mentioned in another answer, the W3C has come up with their own naming system for web colors. I guarantee you will find the same color with 4 different names in those 4 systems. This has not gone unnoticed by others.
One problem with finding a universal set of names, as was attempted by the ISCC-NBS, is that color names may depend on culture and language. This paper summarizes the research on this topic.
"(i) the best examples of basic color categories are the same
within small tolerances of speakers, in any language,
that has the equivalent of the basic color terms in question;
(ii) there is a hierarchy of languages with respect to how
many and which basic color terms they possess (i.e., a
language that has i+ 1 basic color terms features all the
basic terms of any language with i color terms, and any
language with i basic color terms has the same ones);
(iii) basic color categories are characterized by graded
I recommend (1) restricting the size of the problem by restricting the number of colors - use only those colors that are closest to the universal set of colors. This set of 267 colors would be a good starting point to pick a much smaller set of colors. (2) If you have to provide color names to people, use a naming system based on perceptual attributes of the color (here is an example from IBM) rather than names based on cultural references (e.g., indianred) or obscure facts (e.g., burlywood).
Extra Bonus Fact
You can contribute to this research by providing color names of your own.