It's actually Luna Lovegood who discovers it.
Here's how it happens:
* Luna and Harry are looking for a quiet place to practice for their upcoming Defence Against the Dark Arts exam.
* They wander through the castle and stumble upon the Room of Requirement, which Luna, with her unique perspective, recognizes as a place that fulfills the needs of the person seeking it.
Neville does later join Harry and Luna in using the Room of Requirement for practicing, but he doesn't discover it initially. The movie adaptation might have changed this detail, but in the book, Luna is credited with the discovery.