When people give meaningful items to their children, grandchildren, nieces, nephews, and so on, it's called "passing ___ down". People "pass down" things like:

• jewelry
• furniture 
• photographs

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.