The story
Apparently merely from notions such as shaving a little bit off a haggled price: shaving a nickel off the price is then simply demanding or giving a discount. (From that viewpoint, compare take a haircut.) A folk etymology has it that this phrase comes from coin shaving, even though this term and related ones (nickel-shaving and nickel-shaver) apparently did not arise until the 20th century, when coin shaving was already something that was no longer worthwhile; it was still a cultural memory, but it seems less likely as the true origin of this phrase.
Etymology from Wiktionary, CC BY-SA 4.0.