The correct answer is option 1: Karnam Malleswari. Karnam Malleswari is the first Indian woman to win a medal in the Olympics. She achieved this feat in the 2000 Sydney Olympics where she won a bronze medal in weightlifting.

Let`s analyze the other options to understand why they are not the correct answers:

Option 2: Mirabai Chanu. Although Mirabai Chanu is a renowned Indian weightlifter and has won medals in various international competitions, she is not the first Indian woman to win an Olympic medal.

Option 3: P.T. Usha. P.T. Usha is a legendary Indian athlete who has represented India in multiple Olympic Games. However, she has not won a medal in the Olympics, making her the incorrect answer to the question.

Option 4: P.V. Sindhu. P.V. Sindhu is an accomplished Indian badminton player and has won a silver medal in the 2016 Rio Olympics. While she has made India proud with her achievements, she is not the first Indian woman to win an Olympic medal.

Therefore, the correct answer remains option 1: Karnam Malleswari.