"There are gems in the mountain that I too desire. White gems of pure starlight"
Perhaps Thranduil was speaking of the Silmarils, but they weren't made of starlight. Anyone else guess?
Background material for the movies reveal that the White Gems of Lasgalen were brought to the Dwarves to be made into a necklace for Thranduil's queen (though it looks like only some of them were used in the piece). The gems are especially precious to the Elvenking because they remind him of his wife. Keep in mind that this is all original to the films and has only the slightest basis in Tolkien's legendarium where (in The Hobbit) the Elvenking is said to have a fondness for jewels:
If the elf-king had a weakness it was for treasure, especially for silver and white gems; and though his hoard was rich, he was ever eager for more, since he had not yet as great a treasure as other elf-lords of old. His people neither mined nor worked metals or jewels, nor did they bother much with trade or with tilling the earth.
The strange part of this in the films is that Tauriel has no memory of these events, so this would have had to have been several centuries ago, before Tauriel was born or when she was very young. Tauriel is alleged to be about 600 years old. Heck, Thror himself wasn't born until the year 2542 (Third Age); Tauriel was about 200 years older than he was.
If it were up to me, based on Tolkien's essay "Laws and Customs Among the Eldar", I'd retcon Tauriel's age to between 100 and 200 years old. Even 600 years is well-removed from elven childhood.
Not until the fiftieth year did the Eldar attain the stature and shape in which their lives would afterwards endure, and for some a hundred years would pass before they were full-grown.
Interesting thats answered a few questions for