What are some symbolic objects for Thorin?

There are a few things that could be considered symbolic for Thorin: an oak branch, a map and key, and the Arkenstone.

The oak branch is a bit obvious, as it is the source of his epithet "Oakenshield", but it is probably the most important. During the battle he participated in at Moria after his kingdom was taken by Smaug, he broke his shield, so he picked up an oaken branch to use instead, hence the name "Oakenshield".

The map and key are also important to Thorin because they are what started Thorin off on his quest to the Lonely Mountain. Without them, he never would have been able to find and unlock the secret door that would lead them into the mountain.

The final symbol is the Arkenstone, also known as the Heart of the Mountain. It is the symbol of his right to rule, his kingship, which is why he is so eager to find it in The Hobbit: without it, the other dwarves would not follow him.