Option 1: Balasore. Balasore is not well known for the cultivation of coffee. It is a district in Odisha, India, known for its textile and handicraft industries. Coffee cultivation is not a major economic activity in this district.

Option 2: Chikmagalur. Chikmagalur is the correct answer. This district in Karnataka, India, is well known for its coffee plantations. It is often referred to as the "Coffee Land" of Karnataka and is one of the largest coffee producers in the country.

Option 3: Guntur. Guntur is not well known for the cultivation of coffee. It is a district in Andhra Pradesh, India, known for its agricultural produce like chili peppers, cotton, and tobacco. Coffee cultivation is not a major activity in this district.

Option 4: Khurda. Khurda is not well known for the cultivation of coffee. It is a district in Odisha, India, known for its historical and cultural significance. Agriculture is a major occupation in this district, but coffee cultivation is not prominent.

In conclusion, out of the given options, the district well known for the cultivation of coffee is Chikmagalur in Karnataka.