Depends on what you call "original". Is the Servo layout engine _not_ original because layout engines already exist? :)
Moving away from the rhetorical question, I think there are several reasons to rewrite something in Rust, the first and foremost being the huge improvement in memory safety/management. The other reasons are typically related to speed: C wasn't built with concurrency in mind, and trying to develop multi-threaded C applications that run in a cross-platform manner is not straightforward. On the other hand, Rust has the following:
1) Memory safety during concurrency [1]. 2) Multi-threading in the standard library [1]. 3) An amazing package ecosystem revolving around "crates". And there are several build specifically to make concurrency safe and easy to implement. [2]
[1] https://doc.rust-lang.org/stable/book/ch16-00-concurrency.ht... [2] https://crates.io/categories/concurrency