There are a lot of web-based tools that have C implementations, so fun projects may include writing pure-Rust implementations of these. Imagine, say, the CommonMark markdown parser (https://github.com/jgm/CommonMark) implemented in Rust. Making it as fast as the C version would be a challenging optimization problem, but I bet it's doable. :)
I don't necessarily need web-based ideas, but I don't know enough about the systems programming ecosystem to have an idea of where to start! A text editor? A game? A virtual planetarium? Machine learning? I have no idea! (Like I said, not your/Rust's problem...)