Question

What is the name for a process by which voters can propose new laws or change existing ones?

Answer 1

I believe the answer to your question is Initiative. 
I feel this because the definition of an initiative is a process in which citizens can put a proposed new law directly on the ballot in the next election by collecting voters' signatures on a petition. 

Answer 2

Initiative is the faculty or personal quality to use discernment to decide and do things without being told, and take responsibility.

And in politics, Initiative is the process whence a citizen or citizens assume or endorse laws or the pass of new laws or the state constitutional amendments like a referendum.