So according to Biblical texts, the Amalakites were possibly a real historic tribe that living in the area of modern day Israel. The question might be moot depending on historical verification of this group.
Malachite is a green stone, likely used originally for ornamental purposes, but later contributed heavily to the growth of the Copper Age. The name of the stone probably derives from ancient Greek.
So the copper age probably began around 5000 BCE, and biblical stories in Genesis where Amalak (The Amalakites namesake) is written of may have originated as early as 3000 BCE.
Could the name of one have contributed the name of the other? I'm picturing the nomadic Amalakites raiding villages with Copper weapons and taking any Malachite stone they could find. Obviously, that's purely imagination, but could there be some connection?
To the best of my knowledge of both English, Latin, and Hebrew, there is not relation -- the similarity of sounds is purely coincidental. The current name for the naturally-occurring crystalline form of hydrated copper carbonate is from the Greek molochitis, which referrers to the mallow plant (genus Malva). In Greek this is the name not only of a plant, but also to the bright green color of its leaves, sort of the way "orange" in English refers to both a color and a fruit.
In contrast, the Biblical Amalkites are an ethnoym which might mean "dweller of the valley", accoriding to Easton's Bible Dictonary. The Biblical Amalakites were an Idumean people who are speculated to have been ancestral to the Nabateans.