The district famous for the hill temple of Lord Murugan at Palani is:

Madurai
Dindigul
Tiruchirappalli
Theni

The correct answer is Theni.

Palani is a hill town in Theni district in the Indian state of Tamil Nadu. It is one of the six abodes of Murugan, the Hindu god of war and victory. The Palani Murugan Temple is one of the most popular pilgrimage destinations in Tamil Nadu.

Madurai is a city in the Indian state of Tamil Nadu. It is the third-largest city in Tamil Nadu after Chennai and Coimbatore. Madurai is known for its temples, palaces, and museums.

Dindigul is a city in the Indian state of Tamil Nadu. It is the headquarters of Dindigul district. Dindigul is known for its forts, temples, and churches.

Tiruchhirappalli is a city in the Indian state of Tamil Nadu. It is the fourth-largest city in Tamil Nadu after Chennai, Coimbatore, and Madurai. Tiruchirappalli is known for its temples, churches, and mosques.