The Arkenstone of 'The Hobbit' and the Silmarils in 'The Silmarillion,' are keys of difference. The Silmarils, made by waylorn smith Feanor, catch the light from the Two Trees of Valinor, cannot be broken by no power on plain earth. But the Arkenstone was found in earth, can be cut and has no literal glow of divinity. Good legend, too!
They were as tough as plain gravel but gold seemed soft to them, indeed their hard handhold was good only for a little while. The Arkenstone being found in rock and hewn just like any other gem. Also, it should be noted that they say in "The Silmarillion" two of the three brilliant Silmarils ended up settled in places out of reach–in sky or else deep inside planet Earth. And the third one was cast into middle ocean where it vanished forevermore.
So, the result of finding one gem that lay between them in the halls of Erabor is surely questionable. The Arkenstone might have been a different jewel entirely, quite different in beauty and elegance but equally captivating for dwarfs as were the fabled Silmarils.
Discussing Tolkien's gems, aren't they fascinating?! Tempting to imagine that the Arkenstone, the heart of the mountain in 'The Hobbit', could be one of the lost Silmarils from 'The Silmarillion'. That'd be an epic crossover, right? But, I think not, mate. Remember, the three Silmarils were unique, unbreakable jewels crafted by Feanor, containing the divine light of the Two Trees of Valinor. On the contrary, the Arkenstone was unearthed in the Lonely Mountain and was able to be cut, polished, and shaped - which wouldn't be possible if it were a Silmaril. Moreover, according to lore, the Silmarils are meant to be gone from the world, hidden in the sky, the earth's fiery core, and the ocean until the end of days.
So in my opinion, it's unlikely the Arkenstone is a Silmaril, but it's undoubtedly a thematic echo, a kind of homage to the allure and drama of the Silmarils!