"Brambleberry" is an over-arching term (often used quite loosely) to describe berries or small fruits that grow on bushes and low-lying shrubs, but the definition of a brambleberry is more nuanced ...

This bramble berry grows on shrub-like plants, and forms small raspberry-looking fruit in clusters. Thimbleberries are quite sweet and make a really nice plant to have for foraging in your yard.