I would take a pet problem that I have and solve it well. Something so personal that nobody else would solve it. A personal time management app just for me. A way to manage all my contacts so that I can keep up with them intelligently. Something to get me to do that tasks I procrastinate on the worst (like going to the Dentist). There are a million general tools for this, but having your own perfect tool is like having a superpower. And it's a great way to get to know yourself.
Alternatively, you could become the world expert in something small and bizarre. Like VIM. Just kidding. No, like some API. I got to know the MTurk api, and it then became an expert at it. Then people started treating me like an expert. It was cool.
A third approach is find somebody you want to learn from and work with them. Any project you pick will never be as important as what you learn from the project.
Historical curiosity. This answer to "Why did the chicken cross the road" can be interpreted two ways:
1. it's obvious, absurdist humor
2. it is a pun on the phrase "other side" meaning the after life.
Betcha never thought of the second interpretation!