Karnataka is an Indian state. Several etymologies have been suggested for the name Karnataka. The region was popularly referred to as 'Karnata desa' in Indian history. However, historically, the names Karnatak or Carnatic have been misapplied to refer to the regions in or beyond the Western Ghats (Kodagu and Kerala) as well as to a region in present-day Andhra Pradesh (Telangana). The other accepted derivative comes from the Kannada words karu and nādu meaning elevated land, or from the Kannada words kal and nādu meaning rock country, referring to the rocky terrain of the state.
Etymology of Karnataka Wikipedia
Karnataka's name may have originated from the two words Kabbu Nadu, which means land of Sugarcane. This etymology might be possible as Karnataka is a huge Sugarcane-cultivating land. But, this theory might not be very factual as Sugarcane production boomed in South Karnataka only post the building of KRS Dam by Sir M Vishweshwarayya. Before that the very same region of Karnataka was known for Ragi production.