I'm just curious because I always thought an "emerald" was green...
In real life, Emeralds run the gamut from the standard green (the color "emerald", even) to a bluer / yellowish in the extremes, but even then they don't reach the full spectrum of the chaos emeralds.
Putting aside the very nature of them being video game constructs, the fact that the number of emeralds have changed (from 6 to 7), and that their colors have varied, it's otherwise likely that the chaos emeralds aren't really "emeralds" at all, but rather take their name for their association with the Master Emerald, which has always been a large, green gem (and thus, a "real" emerald).
This theory of course falls flat when you consider that the Master Emerald debuted in Sonic & Knuckles, while the Chaos Emeralds have existed since Sonic the Hedgehog (the 1991 game).
Bottom line? Someone in marketing thought "Chaos Emeralds" sounded better than "Chaos Beryls" or "Chaos Diamonds" (which are known for their varied colors) and that choice has propagated ever since, geology be damned.
Because it was called "Chaos Emerald" in the original Japanese.
When Sonic 1 was released in 1991, most Japanese gamers were able to read a small amount of English, but weren't fluent and had only a narrow vocabulary. This allowed game developers to get away with technically inaccurate English. A great example of this is the game Zero Wing, also released in 1991, featuring lines like "All your base are belong to us".
Sonic 1 called the gems Chaos Emeralds (カオスエメラルド - kaosu emerarudo). The American translation kept this name, since it was already in English.
Well, there really isn't any "right" answer to this (unless you happen to know the person who originally designed them), but my guess would be that they're different colors to help distinguish them from each other, as well as the master emerald shards in later games. Think about it, it's more interesting to collect 7 different objects, (even if the differences are cosmetic) than it is to collect 7 identical objects.
One plausible explanation is that they actually meant Beryl, of which Emerald is the green version.
However, if you look at them, the Chaos Emeralds resemble cut diamonds more than anything else.
At a guess, a translator mistranslated it back at Sonic the Hedgehog 1 and the name has stuck ever since.