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Fell Beast! Enjoy being dead
As other people have said, Feanor was kind of childishly eager to prove his merit and status. Maybe that was due to the strong spirit he had or maybe he wanted to show Finwe and Indis that only he was really something.
Miriel had been a very skilled weaver. That might have been the inspiration for Feanor to pursue fame and pleasure from artistry (the writing system he improved and jewellery). Also, Feanor's marriage with a great blacksmith's daughter probably enhanced his interest in jewels.
The main reason for making the silmarils was risen by a question: how to make the light of the trees and the glory of Aman to last forever (The Silmarillion, Ch. 7). To me it seems that Feanor knew that Melkor's kindness was just an act and he had a premonition that something evil was about to happen.
My first thoughts were that Feanor wanted to play a god, a vala, and thus create something that everyone would desire and be amazed at. But the fact that Feanor loathed Melkor had a certain impact, I think. It was only afterwards when he had made the silmarils, that their flawlessness and beauty took over him and he became a bit too fond of them and thus grasping. They were kind of like the One Ring. Feanor didn't want other people to touch the silmarils and was afraid that they would be stolen.
(Scrolls to the beginning to recall what the original question even was...) I agree with Morgul Queen and Sophia about many things but I'd like to think that the idea for the silmarils wasn't completely based on selfish thoughts. I sympathize with Feanor since he must have been very frustrated and unhappy with his life. Though he was ambitious and hot-tempered, he just wanted to make something beautiful endure.