TL;DR: The melange itself is blue. The eye tissue changes color by being saturated with it.
We know that continued use of the substance is required to make the eyes change, so it would appear to be a cumulative effect upon the tissues. Spice is generally thought of as a cinnamon-like substance, probably because the odor is constantly described that way. To make matters more confusing, it was depicted as a reddish-brown powder in both live-action adaptations, as well as the popular video games based on those adaptations.
In "God Emperor of Dune", however, it is described as being more like ground crystal - like sand itself, in fact. As for color, it is a deep blue that refracts light:
The place had filled Moneo with awe. Great bins of melange lay all
around in a gigantic room cut from native rock and illuminated by
glowglobes of an ancient design with arabesques of metal scrollwork
upon them. The spice had glowed radiant blue in the dim silver light.
And the smell-bitter cinnamon, unmistakable.
The color is mentioned again later in that same book (credit to Valorum):
"I, too, am curious. Put your spice-essence on the ledge beside Moneo.
I will keep it."
Slowly, demonstrating by the steadiness of her movements that she
intended no attack, Luyseyal reached beneath her gown and removed a
small vial which glistened with an inner blue radiance. She placed the
vial gently on the ledge. Not by any sign did she indicate that she
might try something desperate.
"Truthsayer, indeed," Leto said.
So apparently the dying of the eyes is literally a saturation of the eye tissue with whatever chemical makes the spice blue.