When people give meaningful items to their children, grandchildren, nieces, nephews, and so on, it's called "passing ___ down". People "pass down" things like:
For things that are valuable but don't have much emotional meaning, we don't use the phrase "pass down". Instead, you say that the relative "gave" or "left" someone these things:
My parents left me a little bit of money when they passed away.
To explain who gets the item, use "to". Use "by" to explain who is giving the item:
It was passed down to me by my mother.