Boyd’s Bible Dictionary:

(fourth ruler). Originally, one governing the fourth part of a country. Under Roman rule, any ruler or petty prince of the republic and empire, especially in Syria (Matt. 14:11At that time Herod the tetrarch heard of the fame of Jesus, (Matthew 14:1); Luke 3:1; 9:71Now in the fifteenth year of the reign of Tiberius Caesar, Pontius Pilate being governor of Judea, and Herod being tetrarch of Galilee, and his brother Philip tetrarch of Iturea and of the region of Trachonitis, and Lysanias the tetrarch of Abilene, (Luke 3:1)
7Now Herod the tetrarch heard of all that was done by him: and he was perplexed, because that it was said of some, that John was risen from the dead; (Luke 9:7)
; Acts 13:11Now there were in the church that was at Antioch certain prophets and teachers; as Barnabas, and Simeon that was called Niger, and Lucius of Cyrene, and Manaen, which had been brought up with Herod the tetrarch, and Saul. (Acts 13:1)). Sometimes called king (Matt. 14:99And the king was sorry: nevertheless for the oath's sake, and them which sat with him at meat, he commanded it to be given her. (Matthew 14:9); Mark 6:14,2214And king Herod heard of him; (for his name was spread abroad:) and he said, That John the Baptist was risen from the dead, and therefore mighty works do show forth themselves in him. (Mark 6:14)
22And when the daughter of the said Herodias came in, and danced, and pleased Herod and them that sat with him, the king said unto the damsel, Ask of me whatsoever thou wilt, and I will give it thee. (Mark 6:22)

Concise Bible Dictionary:

Jackson’s Dictionary of Scripture Proper Names:

ruler of a fourth part (of a country)