It's specifically mentioned in the song as a symbol of William Sycamore's humble beginnings and rustic life.
Here's a passage from the song that illustrates this:
> "He carved his name upon a tree, and drank from a wooden dipper,
> And when he died, the people said, 'He was a man of the people."
This line suggests that William Sycamore was a simple man, content with his life and connected to nature. The wooden dipper represents his simple lifestyle, unburdened by the complexities of wealth and status.