Relation between COLOR and SEX?

Mostly males tend to be more colorful, perhaps its easier to attract a mate if coloration/display shows you to be 'Mr right' for the female.
Good coloration (along with size etc) might tend to indicate robust healthy genetics, in which case the female will choose the healthiest possible partner, in order to pass on that genetic strength/traits to her offspring.
Her main drive is to pass on her genetics, she wants the best. :)

Why it should be color I dont know, In theory it may boost behavioral signalling , sex recognition in habitats of great diversity, where there is much going on, communication may be difficult. Many forest dwellers (males) are especially colorful.
 
Is there one? Vibrancy, brightness, variance, etc?

In most chameleon species yes. If you look at panther chams especially you'll notice the males are the brightly patterned chams most people tend to think about because of their colors. Different regions of Madagascar produce different color patterns, or locales. But the panther females generally all look the same - a pale pinkish grey with some variation. Male veileds are much more colorful than their female counterparts, and their casque on their head is much larger.

Both sexes of carpet chams (furcifer lateralis) are quite colorful, but there is still a big difference between male and female color patterns. The females actually sport more colors in that species. Male brevs (R. brevicaudatus) tend to have more pattern and stripes, but my little female is green and my male is brown. They vary a lot in color so that species isn't very consistent because it may be the other way around too.

Then if you look at Meller chams it's practically impossible to tell them apart. Both sexes look exactly the same.

So it really depends on the species you're considering to answer that question. In general the males are more colorful, which is a common trend in the animal kingdom. Not universal, but more common than the female being more colorful than the male.
 
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