From which goddess name did Hampi get its name?

Pampadevi

The correct answer is Option 4 - Pampadevi

Hampi derived its name from that of the local mother goddess, Pampadevi.

Vijayanagara or “city of victory” was the name of both a city and an empire. The empire was founded in the fourteenth century. In its heyday it stretched from the river Krishna in the north to the extreme south of the peninsula. In 1565 the city was sacked and subsequently deserted. Although it fell into ruin in the seventeenth-eighteenth centuries, it lived on in the memories of people living in the Krishna-Tungabhadra doab. They remembered it as Hampi, a name derived from that of the local mother goddess, Pampadevi. These oral traditions combined with archaeological finds, monuments and inscriptions and other records helped scholars to rediscover the Vijayanagara Empire.