There is zero nickle in a Li-ion cell. Nickel, on the other hand exists in some variants.
Anyway, it looks like the model S uses lithium nickel cobalt aluminium oxide "NCA" LiNiCoAlO2. Aluminium is almost free. lithium compounds are cheap (looks like 5-7k per ton, depending on grade or compound) in comparison to the other metals. Nickel is 18k a ton, and cobalt double that yet.
Both nickel and cobalt weigh around 60g per mole, with lithium a tenth of that, so they also need roughly ten times the mass of nickel and cobalt per mole of LiNiCoAlO2. I guess the lithium is sold as LiOH, which is only roughly a quarter lithium by mass. So to make 1 ton of lithium worth of the compound, you'd need:
4 tons LiOH - $24k
10 tons Ni $180k
10 tons Co $360k
4t aluminium $8k
so the lithium seems to be pretty inconsequential to the overall cost. Mind you those are just metal prices, which I guess is technical grade. If the batteries require better refined metal it probably costs more yet. Then you need graphite for the anode, steel? case, and whatever else bits.
So it looks like the GP is right (barring forgetting cobalt), unless i'm missing something with the chemistry..?